id	sid	tid	token	lemma	pos
ma-117	1	1	2023	2023	NUM
ma-117	1	2	ada	ada	PROPN
ma-117	1	3	academica	academica	PROPN
ma-117	1	4	https://adac.eeeur	https://adac.eeeur	PROPN
ma-117	1	5	.	.	PUNCT
ma-117	2	1	j.	j.	PROPN
ma-117	2	2	math	math	PROPN
ma-117	2	3	.	.	PUNCT
ma-117	3	1	anal	anal	ADJ
ma-117	3	2	.	.	PUNCT
ma-117	4	1	3	3	NUM
ma-117	4	2	(	(	PUNCT
ma-117	4	3	2023	2023	NUM
ma-117	4	4	)	)	PUNCT
ma-117	5	1	8doi	8doi	NUM
ma-117	5	2	:	:	PUNCT
ma-117	5	3	10.28924	10.28924	NUM
ma-117	5	4	/	/	SYM
ma-117	5	5	ada	ada	PROPN
ma-117	5	6	/	/	SYM
ma-117	5	7	ma.3.8	ma.3.8	PROPN
ma-117	5	8	numerical	numerical	ADJ
ma-117	5	9	stabilities	stability	NOUN
ma-117	5	10	of	of	ADP
ma-117	5	11	vasicek	vasicek	NOUN
ma-117	5	12	and	and	CCONJ
ma-117	5	13	geometric	geometric	ADJ
ma-117	5	14	brownian	brownian	ADJ
ma-117	5	15	motion	motion	NOUN
ma-117	5	16	models	model	NOUN
ma-117	5	17	o.	o.	PROPN
ma-117	5	18	c.	c.	PROPN
ma-117	5	19	badibi1,∗	badibi1,∗	PROPN
ma-117	5	20	,	,	PUNCT
ma-117	5	21	i.	i.	NOUN
ma-117	5	22	ramadhani2	ramadhani2	PROPN
ma-117	5	23	,	,	PUNCT
ma-117	5	24	m.	m.	NOUN
ma-117	5	25	a.	a.	PROPN
ma-117	5	26	ndondo1	ndondo1	PROPN
ma-117	5	27	,	,	PUNCT
ma-117	6	1	s.	s.	PROPN
ma-117	6	2	d.	d.	PROPN
ma-117	6	3	kumwimba1	kumwimba1	PROPN
ma-117	7	1	1université	1université	NUM
ma-117	7	2	de	de	X
ma-117	7	3	lubumbashi	lubumbashi	NOUN
ma-117	7	4	,	,	PUNCT
ma-117	7	5	faculté	faculté	PROPN
ma-117	7	6	des	des	PROPN
ma-117	7	7	sciences	sciences	PROPN
ma-117	7	8	,	,	PUNCT
ma-117	7	9	département	département	X
ma-117	7	10	de	de	X
ma-117	7	11	mathématiques	mathématiques	X
ma-117	7	12	et	et	NOUN
ma-117	7	13	informatique	informatique	PROPN
ma-117	7	14	,	,	PUNCT
ma-117	7	15	democratic	democratic	ADJ
ma-117	7	16	republic	republic	NOUN
ma-117	7	17	of	of	ADP
ma-117	7	18	the	the	DET
ma-117	7	19	congo	congo	PROPN
ma-117	7	20	christopheromak2014@gmail.com	christopheromak2014@gmail.com	X
ma-117	7	21	,	,	PUNCT
ma-117	7	22	apondondo@gmail.com	apondondo@gmail.com	X
ma-117	7	23	,	,	PUNCT
ma-117	7	24	didierkumwimba@gmail.com	didierkumwimba@gmail.com	PROPN
ma-117	7	25	2université	2université	NUM
ma-117	7	26	de	de	X
ma-117	7	27	kinshasa	kinshasa	PROPN
ma-117	7	28	,	,	PUNCT
ma-117	7	29	faculté	faculté	PROPN
ma-117	7	30	des	des	PROPN
ma-117	7	31	sciences	sciences	PROPN
ma-117	7	32	et	et	PROPN
ma-117	7	33	technologies	technology	NOUN
ma-117	7	34	,	,	PUNCT
ma-117	7	35	département	département	X
ma-117	7	36	de	de	X
ma-117	7	37	mathématiques	mathématique	NOUN
ma-117	7	38	,	,	PUNCT
ma-117	7	39	informatique	informatique	NOUN
ma-117	7	40	et	et	NOUN
ma-117	7	41	statistiques	statistique	NOUN
ma-117	7	42	,	,	PUNCT
ma-117	7	43	democratic	democratic	ADJ
ma-117	7	44	republic	republic	NOUN
ma-117	7	45	of	of	ADP
ma-117	7	46	the	the	DET
ma-117	7	47	congo	congo	NOUN
ma-117	7	48	issaramadhani@gmail.com	issaramadhani@gmail.com	X
ma-117	7	49	∗correspondence	∗correspondence	NOUN
ma-117	7	50	:	:	PUNCT
ma-117	7	51	christopheromak2014@gmail.com	christopheromak2014@gmail.com	X
ma-117	8	1	abstract	abstract	ADJ
ma-117	8	2	.	.	PUNCT
ma-117	9	1	stochastic	stochastic	ADJ
ma-117	9	2	differential	differential	ADJ
ma-117	9	3	equations	equation	NOUN
ma-117	9	4	(	(	PUNCT
ma-117	9	5	sdes	sde	NOUN
ma-117	9	6	)	)	PUNCT
ma-117	9	7	are	be	AUX
ma-117	9	8	very	very	ADV
ma-117	9	9	often	often	ADV
ma-117	9	10	used	use	VERB
ma-117	9	11	as	as	ADP
ma-117	9	12	models	model	NOUN
ma-117	9	13	for	for	ADP
ma-117	9	14	a	a	DET
ma-117	9	15	large	large	ADJ
ma-117	9	16	numberof	numberof	NOUN
ma-117	9	17	phenomena	phenomena	NOUN
ma-117	9	18	in	in	ADP
ma-117	9	19	the	the	DET
ma-117	9	20	physical	physical	ADJ
ma-117	9	21	,	,	PUNCT
ma-117	9	22	economic	economic	ADJ
ma-117	9	23	and	and	CCONJ
ma-117	9	24	management	management	NOUN
ma-117	9	25	sciences	science	NOUN
ma-117	9	26	.	.	PUNCT
ma-117	10	1	they	they	PRON
ma-117	10	2	generalize	generalize	VERB
ma-117	10	3	the	the	DET
ma-117	10	4	notion	notion	NOUN
ma-117	10	5	ofordinary	ofordinary	ADJ
ma-117	10	6	differential	differential	ADJ
ma-117	10	7	equations	equation	NOUN
ma-117	10	8	,	,	PUNCT
ma-117	10	9	taking	take	VERB
ma-117	10	10	into	into	ADP
ma-117	10	11	account	account	NOUN
ma-117	10	12	a	a	DET
ma-117	10	13	white	white	ADJ
ma-117	10	14	additive	additive	NOUN
ma-117	10	15	and	and	CCONJ
ma-117	10	16	multiplicative	multiplicative	ADJ
ma-117	10	17	noise	noise	NOUN
ma-117	10	18	term	term	NOUN
ma-117	10	19	,	,	PUNCT
ma-117	10	20	tomodel	tomodel	NOUN
ma-117	10	21	random	random	ADJ
ma-117	10	22	trajectories	trajectory	NOUN
ma-117	10	23	such	such	ADJ
ma-117	10	24	as	as	ADP
ma-117	10	25	stock	stock	NOUN
ma-117	10	26	market	market	NOUN
ma-117	10	27	prices	price	NOUN
ma-117	10	28	or	or	CCONJ
ma-117	10	29	particles	particle	NOUN
ma-117	10	30	movements	movement	NOUN
ma-117	10	31	,	,	PUNCT
ma-117	10	32	on	on	ADP
ma-117	10	33	the	the	DET
ma-117	10	34	quantum	quantum	ADJ
ma-117	10	35	scale	scale	NOUN
ma-117	10	36	,	,	PUNCT
ma-117	10	37	subject	subject	ADJ
ma-117	10	38	to	to	ADP
ma-117	10	39	diffusion	diffusion	NOUN
ma-117	10	40	phenomena	phenomenon	NOUN
ma-117	10	41	.	.	PUNCT
ma-117	11	1	in	in	ADP
ma-117	11	2	rare	rare	ADJ
ma-117	11	3	cases	case	NOUN
ma-117	11	4	,	,	PUNCT
ma-117	11	5	it	it	PRON
ma-117	11	6	is	be	AUX
ma-117	11	7	generally	generally	ADV
ma-117	11	8	impossible	impossible	ADJ
ma-117	11	9	to	to	PART
ma-117	11	10	have	have	VERB
ma-117	11	11	explicit	explicit	ADJ
ma-117	11	12	solutionto	solutionto	NOUN
ma-117	11	13	these	these	DET
ma-117	11	14	equations	equation	NOUN
ma-117	11	15	.	.	PUNCT
ma-117	12	1	in	in	ADP
ma-117	12	2	this	this	DET
ma-117	12	3	case	case	NOUN
ma-117	12	4	,	,	PUNCT
ma-117	12	5	the	the	DET
ma-117	12	6	numerical	numerical	ADJ
ma-117	12	7	approach	approach	NOUN
ma-117	12	8	,	,	PUNCT
ma-117	12	9	presenting	present	VERB
ma-117	12	10	itself	itself	PRON
ma-117	12	11	under	under	ADP
ma-117	12	12	various	various	ADJ
ma-117	12	13	aspects	aspect	NOUN
ma-117	12	14	,	,	PUNCT
ma-117	12	15	isthe	isthe	ADJ
ma-117	12	16	only	only	ADJ
ma-117	12	17	favorable	favorable	ADJ
ma-117	12	18	outcome	outcome	NOUN
ma-117	12	19	.	.	PUNCT
ma-117	13	1	however	however	ADV
ma-117	13	2	,	,	PUNCT
ma-117	13	3	the	the	DET
ma-117	13	4	stability	stability	NOUN
ma-117	13	5	of	of	ADP
ma-117	13	6	numerical	numerical	ADJ
ma-117	13	7	schemes	scheme	NOUN
ma-117	13	8	for	for	ADP
ma-117	13	9	stochastic	stochastic	ADJ
ma-117	13	10	differentialequations	differentialequation	NOUN
ma-117	13	11	solution	solution	NOUN
ma-117	13	12	is	be	AUX
ma-117	13	13	much	much	ADV
ma-117	13	14	more	more	ADV
ma-117	13	15	significant	significant	ADJ
ma-117	13	16	.	.	PUNCT
ma-117	14	1	in	in	ADP
ma-117	14	2	this	this	DET
ma-117	14	3	paper	paper	NOUN
ma-117	14	4	,	,	PUNCT
ma-117	14	5	we	we	PRON
ma-117	14	6	establish	establish	VERB
ma-117	14	7	and	and	CCONJ
ma-117	14	8	make	make	VERB
ma-117	14	9	a	a	DET
ma-117	14	10	classical	classical	ADJ
ma-117	14	11	proofof	proofof	NOUN
ma-117	14	12	the	the	DET
ma-117	14	13	mean	mean	ADJ
ma-117	14	14	and	and	CCONJ
ma-117	14	15	mean	mean	ADJ
ma-117	14	16	-	-	PUNCT
ma-117	14	17	square	square	ADJ
ma-117	14	18	stabilities	stability	NOUN
ma-117	14	19	of	of	ADP
ma-117	14	20	the	the	DET
ma-117	14	21	numerical	numerical	ADJ
ma-117	14	22	sdes	sde	NOUN
ma-117	14	23	schemes	scheme	NOUN
ma-117	14	24	for	for	ADP
ma-117	14	25	vasicek	vasicek	NOUN
ma-117	14	26	and	and	CCONJ
ma-117	14	27	geometricbrownian	geometricbrownian	ADJ
ma-117	14	28	motion	motion	NOUN
ma-117	14	29	models	model	NOUN
ma-117	14	30	.	.	PUNCT
ma-117	15	1	1	1	X
ma-117	15	2	.	.	X
ma-117	15	3	introduction	introduction	NOUN
ma-117	15	4	stochastic	stochastic	ADJ
ma-117	15	5	differential	differential	ADJ
ma-117	15	6	equations	equation	NOUN
ma-117	15	7	(	(	PUNCT
ma-117	15	8	sdes	sde	NOUN
ma-117	15	9	)	)	PUNCT
ma-117	15	10	can	can	AUX
ma-117	15	11	be	be	AUX
ma-117	15	12	seen	see	VERB
ma-117	15	13	as	as	ADP
ma-117	15	14	ordinary	ordinary	ADJ
ma-117	15	15	differential	differential	ADJ
ma-117	15	16	equations	equation	NOUN
ma-117	15	17	,	,	PUNCT
ma-117	15	18	or	or	CCONJ
ma-117	15	19	asintegral	asintegral	ADJ
ma-117	15	20	equations	equation	NOUN
ma-117	15	21	in	in	ADP
ma-117	15	22	which	which	PRON
ma-117	15	23	integrals	integral	NOUN
ma-117	15	24	occur	occur	VERB
ma-117	15	25	with	with	ADP
ma-117	15	26	respect	respect	NOUN
ma-117	15	27	to	to	ADP
ma-117	15	28	brownian	brownian	ADJ
ma-117	15	29	motion	motion	NOUN
ma-117	15	30	.	.	PUNCT
ma-117	16	1	they	they	PRON
ma-117	16	2	were	be	AUX
ma-117	16	3	presentedby	presentedby	ADJ
ma-117	16	4	ito	ito	PROPN
ma-117	16	5	,	,	PUNCT
ma-117	16	6	with	with	ADP
ma-117	16	7	the	the	DET
ma-117	16	8	aim	aim	NOUN
ma-117	16	9	of	of	ADP
ma-117	16	10	building	build	VERB
ma-117	16	11	continuous	continuous	ADJ
ma-117	16	12	and	and	CCONJ
ma-117	16	13	strongly	strongly	ADV
ma-117	16	14	markovian	markovian	ADJ
ma-117	16	15	processes	process	NOUN
ma-117	16	16	whose	whose	DET
ma-117	16	17	generatorsare	generatorsare	VERB
ma-117	16	18	second	second	ADJ
ma-117	16	19	-	-	PUNCT
ma-117	16	20	order	order	NOUN
ma-117	16	21	differential	differential	NOUN
ma-117	16	22	operators	operator	NOUN
ma-117	16	23	called	call	VERB
ma-117	16	24	diffusions	diffusion	NOUN
ma-117	16	25	[	[	X
ma-117	16	26	6	6	NUM
ma-117	16	27	]	]	PUNCT
ma-117	16	28	.	.	PUNCT
ma-117	17	1	in	in	ADP
ma-117	17	2	general	general	ADJ
ma-117	17	3	,	,	PUNCT
ma-117	17	4	solving	solve	VERB
ma-117	17	5	explicitly	explicitly	ADV
ma-117	17	6	stochasticdifferential	stochasticdifferential	ADJ
ma-117	17	7	equations	equation	NOUN
ma-117	17	8	(	(	PUNCT
ma-117	17	9	sdes	sde	NOUN
ma-117	17	10	)	)	PUNCT
ma-117	17	11	,	,	PUNCT
ma-117	17	12	except	except	SCONJ
ma-117	17	13	for	for	ADP
ma-117	17	14	cases	case	NOUN
ma-117	17	15	where	where	SCONJ
ma-117	17	16	the	the	DET
ma-117	17	17	diffusion	diffusion	NOUN
ma-117	17	18	and	and	CCONJ
ma-117	17	19	drift	drift	NOUN
ma-117	17	20	coefficients	coefficient	NOUN
ma-117	17	21	are	be	AUX
ma-117	17	22	linears	linear	NOUN
ma-117	17	23	,	,	PUNCT
ma-117	17	24	seems	seem	VERB
ma-117	17	25	difficult	difficult	ADJ
ma-117	17	26	or	or	CCONJ
ma-117	17	27	impossible	impossible	ADJ
ma-117	17	28	[	[	X
ma-117	17	29	8	8	NUM
ma-117	17	30	]	]	PUNCT
ma-117	17	31	.	.	PUNCT
ma-117	18	1	this	this	PRON
ma-117	18	2	is	be	AUX
ma-117	18	3	why	why	SCONJ
ma-117	18	4	the	the	DET
ma-117	18	5	numerical	numerical	ADJ
ma-117	18	6	approach	approach	NOUN
ma-117	18	7	is	be	AUX
ma-117	18	8	relevant	relevant	ADJ
ma-117	18	9	because	because	SCONJ
ma-117	18	10	there	there	PRON
ma-117	18	11	arenumerical	arenumerical	ADJ
ma-117	18	12	methods	method	NOUN
ma-117	18	13	allowing	allow	VERB
ma-117	18	14	to	to	PART
ma-117	18	15	predict	predict	VERB
ma-117	18	16	the	the	DET
ma-117	18	17	qualitative	qualitative	ADJ
ma-117	18	18	behavior	behavior	NOUN
ma-117	18	19	such	such	ADJ
ma-117	18	20	as	as	ADP
ma-117	18	21	the	the	DET
ma-117	18	22	stability	stability	NOUN
ma-117	18	23	of	of	ADP
ma-117	18	24	the	the	DET
ma-117	18	25	solutions	solution	NOUN
ma-117	18	26	.	.	PUNCT
ma-117	19	1	received	receive	VERB
ma-117	19	2	:	:	PUNCT
ma-117	19	3	20	20	NUM
ma-117	19	4	jun	jun	PROPN
ma-117	19	5	2022	2022	NUM
ma-117	19	6	.	.	PUNCT
ma-117	20	1	key	key	ADJ
ma-117	20	2	words	word	NOUN
ma-117	20	3	and	and	CCONJ
ma-117	20	4	phrases	phrase	NOUN
ma-117	20	5	.	.	PUNCT
ma-117	21	1	brownian	brownian	ADJ
ma-117	21	2	motion	motion	NOUN
ma-117	21	3	;	;	PUNCT
ma-117	21	4	stochastic	stochastic	ADJ
ma-117	21	5	differential	differential	ADJ
ma-117	21	6	equations	equation	NOUN
ma-117	21	7	;	;	PUNCT
ma-117	21	8	stabilities	stability	NOUN
ma-117	21	9	of	of	ADP
ma-117	21	10	sdes	sde	NOUN
ma-117	21	11	;	;	PUNCT
ma-117	21	12	numerical	numerical	ADJ
ma-117	21	13	schemes;vasicek	schemes;vasicek	PROPN
ma-117	21	14	and	and	CCONJ
ma-117	21	15	geometric	geometric	ADJ
ma-117	21	16	brownian	brownian	ADJ
ma-117	21	17	motion	motion	NOUN
ma-117	21	18	.	.	PUNCT
ma-117	22	1	1	1	NUM
ma-117	22	2	https://adac.ee	https://adac.ee	PROPN
ma-117	22	3	https://doi.org/10.28924/ada/ma.3.8	https://doi.org/10.28924/ada/ma.3.8	PROPN
ma-117	22	4	eur	eur	NOUN
ma-117	22	5	.	.	PUNCT
ma-117	23	1	j.	j.	PROPN
ma-117	23	2	math	math	PROPN
ma-117	23	3	.	.	PUNCT
ma-117	24	1	anal	anal	PROPN
ma-117	24	2	.	.	PUNCT
ma-117	25	1	10.28924	10.28924	NUM
ma-117	25	2	/	/	SYM
ma-117	25	3	ada	ada	PROPN
ma-117	25	4	/	/	SYM
ma-117	25	5	ma.3.8	ma.3.8	VERB
ma-117	25	6	2the	2the	DET
ma-117	25	7	choice	choice	NOUN
ma-117	25	8	of	of	ADP
ma-117	25	9	a	a	DET
ma-117	25	10	suitable	suitable	ADJ
ma-117	25	11	numerical	numerical	ADJ
ma-117	25	12	scheme	scheme	NOUN
ma-117	25	13	is	be	AUX
ma-117	25	14	based	base	VERB
ma-117	25	15	on	on	ADP
ma-117	25	16	the	the	DET
ma-117	25	17	understanding	understanding	NOUN
ma-117	25	18	and	and	CCONJ
ma-117	25	19	manipulation	manipulation	NOUN
ma-117	25	20	ofcertain	ofcertain	NOUN
ma-117	25	21	qualitative	qualitative	NOUN
ma-117	25	22	properties	property	NOUN
ma-117	25	23	as	as	ADP
ma-117	25	24	stability	stability	NOUN
ma-117	25	25	,	,	PUNCT
ma-117	25	26	consistency	consistency	NOUN
ma-117	25	27	etc	etc	X
ma-117	25	28	.	.	PUNCT
ma-117	26	1	the	the	DET
ma-117	26	2	qualitative	qualitative	ADJ
ma-117	26	3	property	property	NOUN
ma-117	26	4	like	like	ADP
ma-117	26	5	stability	stability	NOUN
ma-117	26	6	ofstochastic	ofstochastic	ADJ
ma-117	26	7	differential	differential	ADJ
ma-117	26	8	equations	equation	NOUN
ma-117	26	9	solutions	solution	NOUN
ma-117	26	10	,	,	PUNCT
ma-117	26	11	introduced	introduce	VERB
ma-117	26	12	by	by	ADP
ma-117	26	13	i.kats	i.kat	NOUN
ma-117	26	14	and	and	CCONJ
ma-117	26	15	n.krasovskii	n.krasovskii	ADJ
ma-117	26	16	[	[	X
ma-117	26	17	2	2	NUM
ma-117	26	18	]	]	PUNCT
ma-117	26	19	and	and	CCONJ
ma-117	26	20	perfectedby	perfectedby	ADJ
ma-117	26	21	i.i	i.i	PROPN
ma-117	26	22	.	.	PROPN
ma-117	26	23	gikhman	gikhman	PROPN
ma-117	26	24	,	,	PUNCT
ma-117	26	25	a.v	a.v	PROPN
ma-117	26	26	.	.	PROPN
ma-117	26	27	skorokhold	skorokhold	PROPN
ma-117	27	1	[	[	X
ma-117	27	2	3	3	X
ma-117	27	3	]	]	PUNCT
ma-117	27	4	and	and	CCONJ
ma-117	27	5	a.	a.	NOUN
ma-117	27	6	friedman	friedman	PROPN
ma-117	27	7	[	[	X
ma-117	27	8	4	4	X
ma-117	27	9	]	]	PUNCT
ma-117	27	10	plays	play	VERB
ma-117	27	11	a	a	DET
ma-117	27	12	major	major	ADJ
ma-117	27	13	role	role	NOUN
ma-117	27	14	in	in	ADP
ma-117	27	15	the	the	DET
ma-117	27	16	study	study	NOUN
ma-117	27	17	of	of	ADP
ma-117	27	18	sdes	sde	NOUN
ma-117	27	19	andthe	andthe	ADJ
ma-117	27	20	numerical	numerical	ADJ
ma-117	27	21	schemes	scheme	NOUN
ma-117	27	22	associated	associate	VERB
ma-117	27	23	.	.	PUNCT
ma-117	28	1	thus	thus	ADV
ma-117	28	2	,	,	PUNCT
ma-117	28	3	looking	look	VERB
ma-117	28	4	for	for	ADP
ma-117	28	5	numerical	numerical	ADJ
ma-117	28	6	schemes	scheme	NOUN
ma-117	28	7	that	that	PRON
ma-117	28	8	preserve	preserve	VERB
ma-117	28	9	qualitativeproperties	qualitativepropertie	NOUN
ma-117	28	10	as	as	SCONJ
ma-117	28	11	the	the	DET
ma-117	28	12	stability	stability	NOUN
ma-117	28	13	of	of	ADP
ma-117	28	14	solutions	solution	NOUN
ma-117	28	15	constitutes	constitute	VERB
ma-117	28	16	and	and	CCONJ
ma-117	28	17	remains	remain	VERB
ma-117	28	18	a	a	DET
ma-117	28	19	very	very	ADV
ma-117	28	20	widespread	widespread	ADJ
ma-117	28	21	problem	problem	NOUN
ma-117	28	22	innumerical	innumerical	ADJ
ma-117	28	23	analysis	analysis	NOUN
ma-117	28	24	of	of	ADP
ma-117	28	25	sdes	sde	NOUN
ma-117	28	26	.	.	PUNCT
ma-117	29	1	in	in	ADP
ma-117	29	2	this	this	DET
ma-117	29	3	article	article	NOUN
ma-117	29	4	we	we	PRON
ma-117	29	5	establish	establish	VERB
ma-117	29	6	and	and	CCONJ
ma-117	29	7	prove	prove	VERB
ma-117	29	8	the	the	DET
ma-117	29	9	conditions	condition	NOUN
ma-117	29	10	of	of	ADP
ma-117	29	11	numerical	numerical	ADJ
ma-117	29	12	schemes	scheme	NOUN
ma-117	29	13	stabilities	stability	NOUN
ma-117	29	14	in	in	ADP
ma-117	29	15	mean	mean	ADJ
ma-117	29	16	andmean	andmean	ADJ
ma-117	29	17	-	-	PUNCT
ma-117	29	18	square	square	NOUN
ma-117	29	19	.	.	PUNCT
ma-117	30	1	we	we	PRON
ma-117	30	2	apply	apply	VERB
ma-117	30	3	the	the	DET
ma-117	30	4	approach	approach	NOUN
ma-117	30	5	described	describe	VERB
ma-117	30	6	by	by	ADP
ma-117	30	7	y.saito	y.saito	ADJ
ma-117	30	8	[	[	X
ma-117	30	9	5	5	NUM
ma-117	30	10	]	]	PUNCT
ma-117	30	11	to	to	PART
ma-117	30	12	defined	define	VERB
ma-117	30	13	and	and	CCONJ
ma-117	30	14	demonstrate	demonstrate	VERB
ma-117	30	15	the	the	DET
ma-117	30	16	sta	sta	NOUN
ma-117	30	17	-	-	PUNCT
ma-117	30	18	bilities	bilitie	NOUN
ma-117	30	19	of	of	ADP
ma-117	30	20	numericals	numerical	NOUN
ma-117	30	21	sdes	sde	VERB
ma-117	30	22	schemes	scheme	NOUN
ma-117	30	23	as	as	ADP
ma-117	30	24	:	:	PUNCT
ma-117	30	25	euler	euler	NOUN
ma-117	30	26	-	-	PUNCT
ma-117	30	27	maruyama	maruyama	NOUN
ma-117	30	28	,	,	PUNCT
ma-117	30	29	milshtein	milshtein	NOUN
ma-117	30	30	and	and	CCONJ
ma-117	30	31	implicit	implicit	ADJ
ma-117	30	32	euler	euler	NOUN
ma-117	30	33	-	-	PUNCT
ma-117	30	34	maruyamafor	maruyamafor	NOUN
ma-117	30	35	vasicek	vasicek	NOUN
ma-117	30	36	and	and	CCONJ
ma-117	30	37	geometric	geometric	ADJ
ma-117	30	38	brownian	brownian	ADJ
ma-117	30	39	motion	motion	NOUN
ma-117	30	40	models	model	NOUN
ma-117	30	41	.	.	PUNCT
ma-117	31	1	to	to	PART
ma-117	31	2	begin	begin	VERB
ma-117	31	3	,	,	PUNCT
ma-117	31	4	let	let	VERB
ma-117	31	5	present	present	VERB
ma-117	31	6	some	some	DET
ma-117	31	7	elementary	elementary	ADJ
ma-117	31	8	notionsrelative	notionsrelative	ADJ
ma-117	31	9	to	to	ADP
ma-117	31	10	sdes	sde	NOUN
ma-117	31	11	and	and	CCONJ
ma-117	31	12	the	the	DET
ma-117	31	13	numerical	numerical	ADJ
ma-117	31	14	schemes	scheme	NOUN
ma-117	31	15	adapted	adapt	VERB
ma-117	31	16	to	to	ADP
ma-117	31	17	the	the	DET
ma-117	31	18	sdes	sde	NOUN
ma-117	31	19	.	.	PUNCT
ma-117	32	1	2	2	X
ma-117	32	2	.	.	X
ma-117	32	3	preliminary	preliminary	ADJ
ma-117	32	4	notions	notion	NOUN
ma-117	32	5	2.1	2.1	NUM
ma-117	32	6	.	.	PUNCT
ma-117	33	1	stochastic	stochastic	ADJ
ma-117	33	2	differential	differential	ADJ
ma-117	33	3	equation	equation	NOUN
ma-117	33	4	and	and	CCONJ
ma-117	33	5	stabilities	stability	NOUN
ma-117	33	6	.	.	PUNCT
ma-117	34	1	in	in	ADP
ma-117	34	2	this	this	DET
ma-117	34	3	section	section	NOUN
ma-117	34	4	,	,	PUNCT
ma-117	34	5	we	we	PRON
ma-117	34	6	present	present	VERB
ma-117	34	7	some	some	DET
ma-117	34	8	definitions	definition	NOUN
ma-117	34	9	inconnection	inconnection	NOUN
ma-117	34	10	with	with	ADP
ma-117	34	11	stochastic	stochastic	ADJ
ma-117	34	12	differential	differential	ADJ
ma-117	34	13	equation	equation	NOUN
ma-117	34	14	and	and	CCONJ
ma-117	34	15	stabilities	stability	NOUN
ma-117	34	16	of	of	ADP
ma-117	34	17	solutions	solution	NOUN
ma-117	34	18	of	of	ADP
ma-117	34	19	sdes	sde	NOUN
ma-117	34	20	.	.	PUNCT
ma-117	35	1	definition	definition	NOUN
ma-117	35	2	2.1	2.1	NUM
ma-117	35	3	.	.	PUNCT
ma-117	36	1	(	(	PUNCT
ma-117	36	2	stochastic	stochastic	ADJ
ma-117	36	3	differential	differential	ADJ
ma-117	36	4	equation	equation	NOUN
ma-117	36	5	(	(	PUNCT
ma-117	36	6	sde	sde	PROPN
ma-117	36	7	)	)	PUNCT
ma-117	37	1	[	[	X
ma-117	37	2	13	13	NUM
ma-117	37	3	]	]	PUNCT
ma-117	37	4	)	)	PUNCT
ma-117	37	5	let	let	VERB
ma-117	37	6	(	(	PUNCT
ma-117	37	7	ω	ω	PROPN
ma-117	37	8	,	,	PUNCT
ma-117	37	9	f	f	PROPN
ma-117	37	10	,	,	PUNCT
ma-117	37	11	(	(	PUNCT
ma-117	37	12	ft)t≥0	ft)t≥0	ADJ
ma-117	37	13	,	,	PUNCT
ma-117	37	14	p	p	NOUN
ma-117	37	15	)	)	PUNCT
ma-117	37	16	be	be	AUX
ma-117	37	17	a	a	DET
ma-117	37	18	filtered	filter	VERB
ma-117	37	19	probability	probability	NOUN
ma-117	37	20	space	space	NOUN
ma-117	37	21	,	,	PUNCT
ma-117	37	22	(	(	PUNCT
ma-117	37	23	bt)t≥0	bt)t≥0	ADP
ma-117	37	24	a	a	DET
ma-117	37	25	standard	standard	ADJ
ma-117	37	26	brownian	brownian	ADJ
ma-117	37	27	motion	motion	NOUN
ma-117	37	28	on	on	ADP
ma-117	37	29	rd	rd	NOUN
ma-117	37	30	defines	define	NOUN
ma-117	37	31	in	in	ADP
ma-117	37	32	a	a	DET
ma-117	37	33	filtered	filter	VERB
ma-117	37	34	probability	probability	NOUN
ma-117	37	35	space	space	NOUN
ma-117	37	36	.	.	PUNCT
ma-117	38	1	a	a	DET
ma-117	38	2	stochastic	stochastic	ADJ
ma-117	38	3	differential	differential	ADJ
ma-117	38	4	equation	equation	NOUN
ma-117	38	5	(	(	PUNCT
ma-117	38	6	sde	sde	PROPN
ma-117	38	7	)	)	PUNCT
ma-117	38	8	on	on	ADP
ma-117	38	9	rd	rd	NOUN
ma-117	38	10	with	with	ADP
ma-117	38	11	the	the	DET
ma-117	38	12	drift	drift	NOUN
ma-117	38	13	coefficient	coefficient	NOUN
ma-117	38	14	:	:	PUNCT
ma-117	38	15	b	b	X
ma-117	38	16	(	(	PUNCT
ma-117	38	17	t	t	PROPN
ma-117	38	18	,	,	PUNCT
ma-117	38	19	xt	xt	X
ma-117	38	20	)	)	PUNCT
ma-117	38	21	∈	∈	PROPN
ma-117	39	1	[	[	X
ma-117	39	2	0	0	NUM
ma-117	39	3	,	,	PUNCT
ma-117	39	4	t	t	X
ma-117	39	5	]	]	X
ma-117	39	6	×	×	PROPN
ma-117	39	7	rn	rn	PROPN
ma-117	39	8	−→	−→	PROPN
ma-117	39	9	rn	rn	PROPN
ma-117	39	10	and	and	CCONJ
ma-117	39	11	the	the	DET
ma-117	39	12	diffusion	diffusion	NOUN
ma-117	39	13	:	:	PUNCT
ma-117	39	14	σ	σ	PROPN
ma-117	39	15	(	(	PUNCT
ma-117	39	16	t	t	PROPN
ma-117	39	17	,	,	PUNCT
ma-117	39	18	xt	xt	X
ma-117	39	19	)	)	PUNCT
ma-117	39	20	∈	∈	PROPN
ma-117	40	1	[	[	X
ma-117	40	2	0	0	NUM
ma-117	40	3	,	,	PUNCT
ma-117	40	4	t	t	X
ma-117	40	5	]	]	X
ma-117	40	6	×	×	PROPN
ma-117	40	7	rn	rn	NOUN
ma-117	40	8	−→	−→	ADJ
ma-117	40	9	rn×d	rn×d	PROPN
ma-117	40	10	when	when	SCONJ
ma-117	40	11	xo	xo	PROPN
ma-117	40	12	is	be	AUX
ma-117	40	13	random	random	ADJ
ma-117	40	14	variable	variable	ADJ
ma-117	40	15	independent	independent	NOUN
ma-117	40	16	of	of	ADP
ma-117	40	17	(	(	PUNCT
ma-117	40	18	bt)t≥0	bt)t≥0	PROPN
ma-117	40	19	is	be	AUX
ma-117	40	20	an	an	DET
ma-117	40	21	equation	equation	NOUN
ma-117	40	22	of	of	ADP
ma-117	40	23	the	the	DET
ma-117	40	24	form	form	NOUN
ma-117	40	25	:	:	PUNCT
ma-117	40	26	{	{	PUNCT
ma-117	40	27	dxt	dxt	PROPN
ma-117	40	28	=	=	SYM
ma-117	40	29	b	b	PROPN
ma-117	40	30	(	(	PUNCT
ma-117	40	31	t	t	PROPN
ma-117	40	32	,	,	PUNCT
ma-117	40	33	xt	xt	X
ma-117	40	34	)	)	PUNCT
ma-117	40	35	dt	dt	PROPN
ma-117	41	1	+	+	CCONJ
ma-117	41	2	σ	σ	PROPN
ma-117	41	3	(	(	PUNCT
ma-117	41	4	t	t	PROPN
ma-117	41	5	,	,	PUNCT
ma-117	41	6	xt	xt	X
ma-117	41	7	)	)	PUNCT
ma-117	41	8	dbt	dbt	PROPN
ma-117	41	9	x	x	SYM
ma-117	41	10	(	(	PUNCT
ma-117	41	11	o	o	NOUN
ma-117	41	12	)	)	PUNCT
ma-117	41	13	=	=	SYM
ma-117	41	14	xo	xo	PROPN
ma-117	41	15	(	(	PUNCT
ma-117	41	16	2.1	2.1	NUM
ma-117	41	17	)	)	PUNCT
ma-117	41	18	the	the	DET
ma-117	41	19	white	white	PROPN
ma-117	41	20	noise	noise	PROPN
ma-117	41	21	σ	σ	PROPN
ma-117	41	22	(	(	PUNCT
ma-117	41	23	t	t	PROPN
ma-117	41	24	,	,	PUNCT
ma-117	41	25	xt	xt	X
ma-117	41	26	)	)	PUNCT
ma-117	41	27	can	can	AUX
ma-117	41	28	be	be	AUX
ma-117	41	29	additive	additive	ADJ
ma-117	41	30	or	or	CCONJ
ma-117	41	31	multiplicative	multiplicative	ADJ
ma-117	41	32	,	,	PUNCT
ma-117	41	33	depending	depend	VERB
ma-117	41	34	on	on	ADP
ma-117	41	35	whether	whether	SCONJ
ma-117	41	36	it	it	PRON
ma-117	41	37	does	do	AUX
ma-117	41	38	notinfluence	notinfluence	NOUN
ma-117	41	39	or	or	CCONJ
ma-117	41	40	does	do	AUX
ma-117	41	41	influence	influence	VERB
ma-117	41	42	the	the	DET
ma-117	41	43	state	state	NOUN
ma-117	41	44	of	of	ADP
ma-117	41	45	the	the	DET
ma-117	41	46	system	system	NOUN
ma-117	41	47	.	.	PUNCT
ma-117	42	1	theorem	theorem	VERB
ma-117	42	2	2.1	2.1	NUM
ma-117	42	3	.	.	PUNCT
ma-117	43	1	(	(	PUNCT
ma-117	43	2	existence	existence	NOUN
ma-117	43	3	and	and	CCONJ
ma-117	43	4	uniqueness	uniqueness	NOUN
ma-117	43	5	[	[	X
ma-117	43	6	14	14	NUM
ma-117	43	7	]	]	PUNCT
ma-117	43	8	)	)	PUNCT
ma-117	43	9	we	we	PRON
ma-117	43	10	assume	assume	VERB
ma-117	43	11	that	that	SCONJ
ma-117	43	12	there	there	PRON
ma-117	43	13	is	be	VERB
ma-117	43	14	a	a	DET
ma-117	43	15	positive	positive	ADJ
ma-117	43	16	constant	constant	ADJ
ma-117	43	17	k	k	NOUN
ma-117	43	18	such	such	ADJ
ma-117	43	19	that	that	DET
ma-117	43	20	∀	∀	NOUN
ma-117	43	21	t	t	NOUN
ma-117	43	22	≥	≥	NOUN
ma-117	43	23	0	0	NUM
ma-117	43	24	,	,	PUNCT
ma-117	43	25	x	x	PRON
ma-117	43	26	,	,	PUNCT
ma-117	43	27	y	y	PROPN
ma-117	43	28	∈	∈	PROPN
ma-117	43	29	rd(1	rd(1	PROPN
ma-117	43	30	)	)	PUNCT
ma-117	43	31	lipschitz	lipschitz	NOUN
ma-117	43	32	condition	condition	NOUN
ma-117	43	33	:	:	PUNCT
ma-117	43	34	|b	|b	PROPN
ma-117	43	35	(	(	PUNCT
ma-117	43	36	t	t	PROPN
ma-117	43	37	,	,	PUNCT
ma-117	43	38	x)−	x)−	PROPN
ma-117	43	39	b	b	PROPN
ma-117	43	40	(	(	PUNCT
ma-117	43	41	t	t	PROPN
ma-117	43	42	,	,	PUNCT
ma-117	43	43	y	y	PROPN
ma-117	43	44	)	)	PUNCT
ma-117	43	45	|+	|+	PROPN
ma-117	43	46	|σ	|σ	X
ma-117	43	47	(	(	PUNCT
ma-117	43	48	t	t	PROPN
ma-117	43	49	,	,	PUNCT
ma-117	43	50	x)−	x)−	PROPN
ma-117	43	51	σ	σ	PROPN
ma-117	43	52	(	(	PUNCT
ma-117	43	53	t	t	PROPN
ma-117	43	54	,	,	PUNCT
ma-117	43	55	y	y	PROPN
ma-117	43	56	)	)	PUNCT
ma-117	43	57	|	|	ADV
ma-117	43	58	≤	≤	NUM
ma-117	43	59	k|x	k|x	NOUN
ma-117	44	1	−	−	NOUN
ma-117	44	2	y	y	PROPN
ma-117	45	1	|	|	ADV
ma-117	45	2	https://doi.org/10.28924/ada/ma.3.8	https://doi.org/10.28924/ada/ma.3.8	PROPN
ma-117	45	3	eur	eur	PROPN
ma-117	45	4	.	.	PUNCT
ma-117	46	1	j.	j.	PROPN
ma-117	46	2	math	math	PROPN
ma-117	46	3	.	.	PUNCT
ma-117	47	1	anal	anal	PROPN
ma-117	47	2	.	.	PUNCT
ma-117	48	1	10.28924	10.28924	NUM
ma-117	48	2	/	/	SYM
ma-117	48	3	ada	ada	PROPN
ma-117	48	4	/	/	SYM
ma-117	48	5	ma.3.8	ma.3.8	PROPN
ma-117	48	6	3(2	3(2	NUM
ma-117	48	7	)	)	PUNCT
ma-117	48	8	linear	linear	ADJ
ma-117	48	9	growth	growth	NOUN
ma-117	48	10	condition	condition	NOUN
ma-117	48	11	:	:	PUNCT
ma-117	48	12	|b	|b	PROPN
ma-117	48	13	(	(	PUNCT
ma-117	48	14	t	t	PROPN
ma-117	48	15	,	,	PUNCT
ma-117	48	16	x	x	NOUN
ma-117	48	17	)	)	PUNCT
ma-117	49	1	|	|	ADV
ma-117	49	2	≤	≤	ADV
ma-117	49	3	k	k	X
ma-117	49	4	(	(	PUNCT
ma-117	49	5	1	1	NUM
ma-117	49	6	+	+	NUM
ma-117	49	7	|x|	|x|	PROPN
ma-117	49	8	)	)	PUNCT
ma-117	49	9	,	,	PUNCT
ma-117	49	10	|σ	|σ	X
ma-117	49	11	(	(	PUNCT
ma-117	49	12	t	t	PROPN
ma-117	49	13	,	,	PUNCT
ma-117	49	14	x	x	NOUN
ma-117	49	15	)	)	PUNCT
ma-117	49	16	|	|	ADV
ma-117	49	17	≤	≤	ADV
ma-117	49	18	k	k	X
ma-117	49	19	(	(	PUNCT
ma-117	49	20	1	1	NUM
ma-117	49	21	+	+	NUM
ma-117	49	22	|x|	|x|	PROPN
ma-117	49	23	)	)	PUNCT
ma-117	49	24	so	so	SCONJ
ma-117	49	25	the	the	DET
ma-117	49	26	sde	sde	PROPN
ma-117	49	27	(	(	PUNCT
ma-117	49	28	2.1	2.1	NUM
ma-117	49	29	)	)	PUNCT
ma-117	49	30	admits	admit	VERB
ma-117	49	31	,	,	PUNCT
ma-117	49	32	for	for	SCONJ
ma-117	49	33	any	any	DET
ma-117	49	34	initial	initial	ADJ
ma-117	49	35	condition	condition	NOUN
ma-117	49	36	xo	xo	PROPN
ma-117	49	37	of	of	ADP
ma-117	49	38	square	square	PROPN
ma-117	49	39	integrable	integrable	ADJ
ma-117	49	40	(	(	PUNCT
ma-117	49	41	e	e	X
ma-117	49	42	[	[	PUNCT
ma-117	49	43	|xo	|xo	X
ma-117	49	44	|2	|2	NUM
ma-117	49	45	]	]	PUNCT
ma-117	50	1	<	<	X
ma-117	50	2	∞	∞	NUM
ma-117	50	3	)	)	PUNCT
ma-117	50	4	the	the	DET
ma-117	50	5	strong	strong	ADJ
ma-117	50	6	solution	solution	NOUN
ma-117	50	7	(	(	PUNCT
ma-117	50	8	xt)t∈[0,t	xt)t∈[0,t	NOUN
ma-117	50	9	]	]	X
ma-117	50	10	,	,	PUNCT
ma-117	50	11	unique	unique	ADJ
ma-117	50	12	,	,	PUNCT
ma-117	50	13	almost	almost	ADV
ma-117	50	14	surely	surely	ADV
ma-117	50	15	continuous	continuous	ADJ
ma-117	50	16	and	and	CCONJ
ma-117	50	17	satisfying	satisfy	VERB
ma-117	50	18	the	the	DET
ma-117	50	19	following	follow	VERB
ma-117	50	20	condition	condition	NOUN
ma-117	50	21	:	:	PUNCT
ma-117	50	22	e	e	X
ma-117	50	23	(	(	PUNCT
ma-117	50	24	sup	sup	PROPN
ma-117	50	25	0≤t≤t	0≤t≤t	NUM
ma-117	50	26	|x2	|x2	NOUN
ma-117	50	27	t	t	NOUN
ma-117	50	28	|	|	ADV
ma-117	50	29	)	)	PUNCT
ma-117	51	1	<	<	X
ma-117	51	2	∞	∞	NUM
ma-117	51	3	definition	definition	NOUN
ma-117	51	4	2.2	2.2	NUM
ma-117	51	5	.	.	PUNCT
ma-117	52	1	(	(	PUNCT
ma-117	52	2	asymptotic	asymptotic	ADJ
ma-117	52	3	stability	stability	NOUN
ma-117	52	4	in	in	ADP
ma-117	52	5	probability	probability	NOUN
ma-117	52	6	in	in	ADP
ma-117	52	7	large	large	ADJ
ma-117	52	8	sense	sense	NOUN
ma-117	52	9	[	[	X
ma-117	52	10	1	1	NUM
ma-117	52	11	]	]	PUNCT
ma-117	52	12	,	,	PUNCT
ma-117	52	13	[	[	X
ma-117	52	14	24	24	NUM
ma-117	52	15	]	]	PUNCT
ma-117	52	16	)	)	PUNCT
ma-117	52	17	the	the	DET
ma-117	52	18	solution	solution	NOUN
ma-117	52	19	is	be	AUX
ma-117	52	20	said	say	VERB
ma-117	52	21	to	to	PART
ma-117	52	22	be	be	AUX
ma-117	52	23	asymptotically	asymptotically	ADV
ma-117	52	24	and	and	CCONJ
ma-117	52	25	stochastically	stochastically	ADV
ma-117	52	26	stable	stable	ADJ
ma-117	52	27	in	in	ADP
ma-117	52	28	the	the	DET
ma-117	52	29	large	large	ADJ
ma-117	52	30	sense	sense	NOUN
ma-117	52	31	if	if	SCONJ
ma-117	52	32	∀	∀	NUM
ma-117	52	33	xo	xo	PROPN
ma-117	52	34	∈	∈	PROPN
ma-117	52	35	l2	l2	NOUN
ma-117	52	36	ft	ft	X
ma-117	52	37	(	(	PUNCT
ma-117	52	38	[	[	X
ma-117	52	39	−t	−t	NOUN
ma-117	52	40	,	,	PUNCT
ma-117	52	41	0	0	NUM
ma-117	52	42	]	]	PUNCT
ma-117	52	43	,	,	PUNCT
ma-117	52	44	rn	rn	PROPN
ma-117	52	45	)	)	PUNCT
ma-117	52	46	,	,	PUNCT
ma-117	52	47	then	then	ADV
ma-117	52	48	p	p	X
ma-117	52	49	{	{	PUNCT
ma-117	52	50	lim	lim	PROPN
ma-117	52	51	t−→∞	t−→∞	PROPN
ma-117	52	52	x	x	SYM
ma-117	52	53	(	(	PUNCT
ma-117	52	54	t	t	NOUN
ma-117	52	55	)	)	PUNCT
ma-117	52	56	=	=	SYM
ma-117	52	57	0	0	PUNCT
ma-117	53	1	}	}	PUNCT
ma-117	53	2	=	=	SYM
ma-117	53	3	1	1	X
ma-117	53	4	.	.	PUNCT
ma-117	53	5	definition	definition	NOUN
ma-117	53	6	2.3	2.3	NUM
ma-117	53	7	.	.	PUNCT
ma-117	54	1	(	(	PUNCT
ma-117	54	2	stability	stability	NOUN
ma-117	54	3	of	of	ADP
ma-117	54	4	pth	pth	NOUN
ma-117	54	5	moment	moment	NOUN
ma-117	55	1	[	[	X
ma-117	55	2	23	23	NUM
ma-117	55	3	]	]	PUNCT
ma-117	55	4	,	,	PUNCT
ma-117	55	5	[	[	X
ma-117	55	6	25])(1	25])(1	NOUN
ma-117	55	7	)	)	PUNCT
ma-117	55	8	let	let	VERB
ma-117	55	9	p	p	PRON
ma-117	55	10	≥	≥	NUM
ma-117	55	11	2	2	NUM
ma-117	55	12	we	we	PRON
ma-117	55	13	say	say	VERB
ma-117	55	14	that	that	SCONJ
ma-117	55	15	a	a	DET
ma-117	55	16	solution	solution	NOUN
ma-117	55	17	of	of	ADP
ma-117	55	18	(	(	PUNCT
ma-117	55	19	2.1	2.1	NUM
ma-117	55	20	)	)	PUNCT
ma-117	55	21	is	be	AUX
ma-117	55	22	stable	stable	ADJ
ma-117	55	23	in	in	ADP
ma-117	55	24	pth	pth	NOUN
ma-117	55	25	moment	moment	NOUN
ma-117	55	26	if	if	SCONJ
ma-117	55	27	∀ε	∀ε	PROPN
ma-117	55	28	>	>	X
ma-117	55	29	0	0	NUM
ma-117	56	1	it	it	PRON
ma-117	56	2	exists	exist	VERB
ma-117	56	3	δ	δ	PROPN
ma-117	56	4	>	>	X
ma-117	56	5	0	0	NUM
ma-117	56	6	such	such	ADJ
ma-117	56	7	as	as	ADP
ma-117	56	8	e	e	NOUN
ma-117	56	9	[	[	PUNCT
ma-117	56	10	sup	sup	NOUN
ma-117	56	11	t>0	t>0	NOUN
ma-117	56	12	|x	|x	NOUN
ma-117	56	13	(	(	PUNCT
ma-117	56	14	t	t	NOUN
ma-117	56	15	)	)	PUNCT
ma-117	56	16	|p	|p	NOUN
ma-117	56	17	]	]	PUNCT
ma-117	56	18	<	<	X
ma-117	56	19	ε	ε	X
ma-117	56	20	avec	avec	X
ma-117	56	21	|xo	|xo	PROPN
ma-117	57	1	|	|	ADV
ma-117	57	2	<	<	X
ma-117	57	3	δ	δ	PROPN
ma-117	57	4	(	(	PUNCT
ma-117	57	5	2	2	X
ma-117	57	6	)	)	PUNCT
ma-117	57	7	let	let	VERB
ma-117	57	8	p	p	PRON
ma-117	57	9	≥	≥	NOUN
ma-117	57	10	2	2	NUM
ma-117	57	11	,	,	PUNCT
ma-117	57	12	we	we	PRON
ma-117	57	13	say	say	VERB
ma-117	57	14	that	that	SCONJ
ma-117	57	15	a	a	DET
ma-117	57	16	solution	solution	NOUN
ma-117	57	17	of	of	ADP
ma-117	57	18	(	(	PUNCT
ma-117	57	19	2.1	2.1	NUM
ma-117	57	20	)	)	PUNCT
ma-117	57	21	is	be	AUX
ma-117	57	22	stable	stable	ADJ
ma-117	57	23	asymptoticaly	asymptoticaly	NOUN
ma-117	57	24	in	in	ADP
ma-117	57	25	pth	pth	NOUN
ma-117	57	26	moment	moment	NOUN
ma-117	57	27	if	if	SCONJ
ma-117	57	28	it	it	PRON
ma-117	57	29	is	be	AUX
ma-117	57	30	stable	stable	ADJ
ma-117	57	31	from	from	ADP
ma-117	57	32	peme	peme	ADJ
ma-117	57	33	moment	moment	NOUN
ma-117	57	34	∀	∀	NOUN
ma-117	57	35	xo	xo	PROPN
ma-117	57	36	∈	∈	PROPN
ma-117	57	37	l2	l2	NOUN
ma-117	57	38	fto	fto	NOUN
ma-117	57	39	(	(	PUNCT
ma-117	57	40	[	[	X
ma-117	57	41	−t	−t	NOUN
ma-117	57	42	,	,	PUNCT
ma-117	57	43	0	0	NUM
ma-117	57	44	]	]	PUNCT
ma-117	57	45	,	,	PUNCT
ma-117	57	46	rn	rn	PROPN
ma-117	57	47	)	)	PUNCT
ma-117	57	48	then	then	ADV
ma-117	57	49	we	we	PRON
ma-117	57	50	have	have	VERB
ma-117	57	51	:	:	PUNCT
ma-117	57	52	lim	lim	PROPN
ma-117	57	53	t−→∞	t−→∞	PROPN
ma-117	57	54	e	e	PROPN
ma-117	57	55	[	[	PUNCT
ma-117	57	56	sup	sup	NOUN
ma-117	57	57	t	t	PROPN
ma-117	57	58	>	>	X
ma-117	57	59	t	t	PROPN
ma-117	57	60	|x	|x	NOUN
ma-117	57	61	(	(	PUNCT
ma-117	57	62	t	t	NOUN
ma-117	57	63	)	)	PUNCT
ma-117	57	64	|p	|p	NOUN
ma-117	57	65	]	]	PUNCT
ma-117	57	66	=	=	SYM
ma-117	57	67	0	0	NUM
ma-117	57	68	2.2	2.2	NUM
ma-117	57	69	.	.	PUNCT
ma-117	58	1	stochastic	stochastic	ADJ
ma-117	58	2	numerical	numerical	ADJ
ma-117	58	3	schemes	scheme	NOUN
ma-117	58	4	.	.	PUNCT
ma-117	59	1	in	in	ADP
ma-117	59	2	this	this	DET
ma-117	59	3	section	section	NOUN
ma-117	59	4	we	we	PRON
ma-117	59	5	present	present	VERB
ma-117	59	6	three	three	NUM
ma-117	59	7	numerical	numerical	ADJ
ma-117	59	8	schemes	scheme	NOUN
ma-117	59	9	as	as	ADP
ma-117	59	10	euler	euler	NOUN
ma-117	59	11	-	-	PUNCT
ma-117	59	12	maruyama	maruyama	NOUN
ma-117	59	13	,	,	PUNCT
ma-117	59	14	implicit	implicit	ADJ
ma-117	59	15	euler	euler	NOUN
ma-117	59	16	-	-	PUNCT
ma-117	59	17	maruyama	maruyama	NOUN
ma-117	59	18	and	and	CCONJ
ma-117	59	19	milshtein	milshtein	PROPN
ma-117	59	20	schemes	scheme	NOUN
ma-117	59	21	.	.	PUNCT
ma-117	60	1	definition	definition	NOUN
ma-117	60	2	2.4	2.4	NUM
ma-117	60	3	.	.	PUNCT
ma-117	61	1	(	(	PUNCT
ma-117	61	2	euler	euler	NOUN
ma-117	61	3	-	-	PUNCT
ma-117	61	4	maruyama	maruyama	NOUN
ma-117	61	5	scheme	scheme	NOUN
ma-117	61	6	[	[	X
ma-117	61	7	10	10	NUM
ma-117	61	8	]	]	PUNCT
ma-117	61	9	,	,	PUNCT
ma-117	62	1	[	[	X
ma-117	62	2	11	11	NUM
ma-117	62	3	]	]	PUNCT
ma-117	62	4	)	)	PUNCT
ma-117	62	5	let	let	VERB
ma-117	62	6	{	{	PUNCT
ma-117	62	7	xt	xt	ADP
ma-117	62	8	}	}	PUNCT
ma-117	62	9	the	the	DET
ma-117	62	10	diffusion	diffusion	NOUN
ma-117	62	11	solution	solution	NOUN
ma-117	62	12	of	of	ADP
ma-117	62	13	the	the	DET
ma-117	62	14	sde(2.1	sde(2.1	NUM
ma-117	62	15	)	)	PUNCT
ma-117	62	16	.	.	PUNCT
ma-117	63	1	let	let	AUX
ma-117	63	2	consider	consider	VERB
ma-117	63	3	the	the	DET
ma-117	63	4	interval	interval	NOUN
ma-117	63	5	[	[	X
ma-117	63	6	0	0	NUM
ma-117	63	7	,	,	PUNCT
ma-117	63	8	t	t	NOUN
ma-117	63	9	]	]	PUNCT
ma-117	63	10	and	and	CCONJ
ma-117	63	11	a	a	DET
ma-117	63	12	regular	regular	ADJ
ma-117	63	13	subdivision	subdivision	NOUN
ma-117	63	14	t0	t0	NOUN
ma-117	63	15	=	=	SYM
ma-117	63	16	0	0	PUNCT
ma-117	63	17	<	<	X
ma-117	63	18	t1	t1	NOUN
ma-117	63	19	<	<	X
ma-117	63	20	t2	t2	PROPN
ma-117	63	21	<	<	X
ma-117	63	22	t0	t0	X
ma-117	63	23	<	<	X
ma-117	63	24	·	·	PUNCT
ma-117	63	25	·	·	PUNCT
ma-117	63	26	·	·	PUNCT
ma-117	64	1	<	<	X
ma-117	64	2	tk	tk	PROPN
ma-117	64	3	=	=	PROPN
ma-117	64	4	t	t	PROPN
ma-117	64	5	with	with	ADP
ma-117	64	6	step	step	NOUN
ma-117	64	7	∆t	∆t	PROPN
ma-117	64	8	=	=	SYM
ma-117	64	9	t	t	PROPN
ma-117	64	10	n	n	NOUN
ma-117	64	11	=	=	SYM
ma-117	64	12	t	t	PROPN
ma-117	64	13	k	k	PROPN
ma-117	64	14	,	,	PUNCT
ma-117	64	15	the	the	DET
ma-117	64	16	euler	euler	NOUN
ma-117	64	17	-	-	PUNCT
ma-117	64	18	maruyama	maruyama	NOUN
ma-117	64	19	scheme	scheme	NOUN
ma-117	64	20	of	of	ADP
ma-117	64	21	(	(	PUNCT
ma-117	64	22	2.1	2.1	NUM
ma-117	64	23	)	)	PUNCT
ma-117	64	24	is	be	AUX
ma-117	64	25	defined	define	VERB
ma-117	64	26	like	like	ADP
ma-117	64	27	:	:	PUNCT
ma-117	64	28	{	{	PUNCT
ma-117	64	29	xemk+1	xemk+1	PROPN
ma-117	64	30	=	=	SYM
ma-117	64	31	xk	xk	PROPN
ma-117	64	32	+	+	CCONJ
ma-117	64	33	b(tk	b(tk	PROPN
ma-117	64	34	,	,	PUNCT
ma-117	64	35	xk)(tk+1	xk)(tk+1	PROPN
ma-117	65	1	−	−	PROPN
ma-117	65	2	tk	tk	PROPN
ma-117	65	3	)	)	PUNCT
ma-117	65	4	+	+	NUM
ma-117	65	5	σ(tk	σ(tk	NOUN
ma-117	65	6	,	,	PUNCT
ma-117	65	7	xk)(bk+1	xk)(bk+1	PUNCT
ma-117	66	1	−	−	PUNCT
ma-117	67	1	bk	bk	NOUN
ma-117	67	2	)	)	PUNCT
ma-117	67	3	x(0	x(0	PROPN
ma-117	67	4	)	)	PUNCT
ma-117	68	1	=	=	SYM
ma-117	69	1	x0	x0	PROPN
ma-117	69	2	(	(	PUNCT
ma-117	69	3	2.2	2.2	NUM
ma-117	69	4	)	)	PUNCT
ma-117	69	5	https://doi.org/10.28924/ada/ma.3.8	https://doi.org/10.28924/ada/ma.3.8	NUM
ma-117	69	6	eur	eur	NOUN
ma-117	69	7	.	.	PUNCT
ma-117	70	1	j.	j.	PROPN
ma-117	70	2	math	math	PROPN
ma-117	70	3	.	.	PUNCT
ma-117	71	1	anal	anal	PROPN
ma-117	71	2	.	.	PUNCT
ma-117	72	1	10.28924	10.28924	NUM
ma-117	72	2	/	/	SYM
ma-117	72	3	ada	ada	PROPN
ma-117	72	4	/	/	SYM
ma-117	72	5	ma.3.8	ma.3.8	PROPN
ma-117	72	6	4	4	NUM
ma-117	72	7	definition	definition	NOUN
ma-117	72	8	2.5	2.5	NUM
ma-117	72	9	.	.	PUNCT
ma-117	73	1	(	(	PUNCT
ma-117	73	2	implicit	implicit	ADJ
ma-117	73	3	euler	euler	NOUN
ma-117	73	4	-	-	PUNCT
ma-117	73	5	maruyama	maruyama	NOUN
ma-117	73	6	scheme	scheme	NOUN
ma-117	73	7	[	[	X
ma-117	73	8	10	10	NUM
ma-117	73	9	]	]	PUNCT
ma-117	73	10	)	)	PUNCT
ma-117	73	11	the	the	DET
ma-117	73	12	implicit	implicit	ADJ
ma-117	73	13	euler	euler	NOUN
ma-117	73	14	-	-	PUNCT
ma-117	73	15	maruyama	maruyama	NOUN
ma-117	73	16	scheme	scheme	NOUN
ma-117	73	17	is	be	AUX
ma-117	73	18	a	a	DET
ma-117	73	19	convergent	convergent	NOUN
ma-117	73	20	scheme	scheme	NOUN
ma-117	73	21	like	like	ADP
ma-117	73	22	the	the	DET
ma-117	73	23	euler	euler	NOUN
ma-117	73	24	-	-	PUNCT
ma-117	73	25	maruyama	maruyama	NOUN
ma-117	73	26	scheme	scheme	NOUN
ma-117	73	27	.	.	PUNCT
ma-117	74	1	to	to	PART
ma-117	74	2	be	be	AUX
ma-117	74	3	reassured	reassure	VERB
ma-117	74	4	of	of	ADP
ma-117	74	5	the	the	DET
ma-117	74	6	existence	existence	NOUN
ma-117	74	7	of	of	ADP
ma-117	74	8	the	the	DET
ma-117	74	9	solutions	solution	NOUN
ma-117	74	10	of	of	ADP
ma-117	74	11	this	this	DET
ma-117	74	12	scheme	scheme	NOUN
ma-117	74	13	,	,	PUNCT
ma-117	74	14	only	only	ADV
ma-117	74	15	the	the	DET
ma-117	74	16	term	term	NOUN
ma-117	74	17	of	of	ADP
ma-117	74	18	the	the	DET
ma-117	74	19	drift	drift	NOUN
ma-117	74	20	is	be	AUX
ma-117	74	21	implicit	implicit	ADJ
ma-117	74	22	.	.	PUNCT
ma-117	75	1	for	for	ADP
ma-117	75	2	this	this	DET
ma-117	75	3	fact	fact	NOUN
ma-117	75	4	:	:	PUNCT
ma-117	75	5	b(xk)∆tk	b(xk)∆tk	ADJ
ma-117	75	6	which	which	PRON
ma-117	75	7	is	be	AUX
ma-117	75	8	in	in	ADP
ma-117	75	9	the	the	DET
ma-117	75	10	euler	euler	NOUN
ma-117	75	11	-	-	PUNCT
ma-117	75	12	maruyama	maruyama	NOUN
ma-117	75	13	scheme	scheme	NOUN
ma-117	75	14	is	be	AUX
ma-117	75	15	replaced	replace	VERB
ma-117	75	16	by	by	ADP
ma-117	75	17	b(xk+1)∆tk	b(xk+1)∆tk	PROPN
ma-117	75	18	and	and	CCONJ
ma-117	75	19	the	the	DET
ma-117	75	20	diffusion	diffusion	NOUN
ma-117	75	21	term	term	NOUN
ma-117	75	22	:	:	PUNCT
ma-117	75	23	σ(xk)∆bk	σ(xk)∆bk	NOUN
ma-117	75	24	remains	remain	VERB
ma-117	75	25	unchanged	unchanged	ADJ
ma-117	75	26	.	.	PUNCT
ma-117	76	1	the	the	DET
ma-117	76	2	implicit	implicit	ADJ
ma-117	76	3	euler	euler	VERB
ma-117	76	4	-	-	PUNCT
ma-117	76	5	maruyama	maruyama	NOUN
ma-117	76	6	scheme	scheme	NOUN
ma-117	76	7	of	of	ADP
ma-117	76	8	the	the	DET
ma-117	76	9	eds	ed	NOUN
ma-117	76	10	(	(	PUNCT
ma-117	76	11	2.1	2.1	NUM
ma-117	76	12	)	)	PUNCT
ma-117	76	13	has	have	AUX
ma-117	76	14	given	give	VERB
ma-117	76	15	by	by	ADP
ma-117	76	16	:	:	PUNCT
ma-117	76	17	xiemk+1	xiemk+1	PROPN
ma-117	76	18	=	=	SYM
ma-117	76	19	xk	xk	PROPN
ma-117	76	20	+	+	CCONJ
ma-117	76	21	b(xk+1)∆tk	b(xk+1)∆tk	PROPN
ma-117	76	22	+	+	CCONJ
ma-117	76	23	σ(xk)∆bk	σ(xk)∆bk	NOUN
ma-117	76	24	(	(	PUNCT
ma-117	76	25	2.3	2.3	NUM
ma-117	76	26	)	)	PUNCT
ma-117	76	27	definition	definition	NOUN
ma-117	76	28	2.6	2.6	NUM
ma-117	76	29	.	.	PUNCT
ma-117	77	1	(	(	PUNCT
ma-117	77	2	milshtein	milshtein	NOUN
ma-117	77	3	scheme	scheme	NOUN
ma-117	77	4	[	[	X
ma-117	77	5	7	7	NUM
ma-117	77	6	]	]	PUNCT
ma-117	77	7	)	)	PUNCT
ma-117	77	8	let	let	AUX
ma-117	77	9	consider	consider	VERB
ma-117	77	10	the	the	DET
ma-117	77	11	sde	sde	PROPN
ma-117	77	12	(	(	PUNCT
ma-117	77	13	2.1	2.1	NUM
ma-117	77	14	)	)	PUNCT
ma-117	77	15	and	and	CCONJ
ma-117	77	16	a	a	DET
ma-117	77	17	regular	regular	ADJ
ma-117	77	18	subdivision	subdivision	NOUN
ma-117	77	19	of	of	ADP
ma-117	77	20	the	the	DET
ma-117	77	21	intervalle	intervalle	PROPN
ma-117	77	22	et	et	PROPN
ma-117	77	23	une	une	NOUN
ma-117	77	24	subdivision	subdivision	NOUN
ma-117	77	25	of	of	ADP
ma-117	77	26	the	the	DET
ma-117	77	27	interval	interval	NOUN
ma-117	77	28	[	[	X
ma-117	77	29	0	0	NUM
ma-117	77	30	,	,	PUNCT
ma-117	77	31	t	t	X
ma-117	77	32	]	]	X
ma-117	77	33	:	:	PUNCT
ma-117	77	34	0	0	X
ma-117	78	1	=	=	SYM
ma-117	78	2	t0	t0	PROPN
ma-117	78	3	<	<	X
ma-117	78	4	t1	t1	NOUN
ma-117	78	5	<	<	X
ma-117	78	6	t2	t2	PROPN
ma-117	78	7	<	<	X
ma-117	78	8	·	·	PUNCT
ma-117	78	9	·	·	PUNCT
ma-117	78	10	·	·	PUNCT
ma-117	79	1	<	<	X
ma-117	79	2	tn	tn	PROPN
ma-117	79	3	=	=	SYM
ma-117	79	4	t	t	X
ma-117	79	5	de	de	X
ma-117	80	1	[	[	X
ma-117	80	2	0	0	NUM
ma-117	80	3	,	,	PUNCT
ma-117	80	4	t	t	X
ma-117	80	5	]	]	PUNCT
ma-117	80	6	the	the	DET
ma-117	80	7	milshtein	milshtein	PROPN
ma-117	80	8	scheme	scheme	NOUN
ma-117	80	9	is	be	AUX
ma-117	80	10	defined	define	VERB
ma-117	80	11	like:	like:	NOUN
ma-117	80	12	xmk+1	xmk+1	PUNCT
ma-117	81	1	=	=	PUNCT
ma-117	81	2	xk	xk	PROPN
ma-117	82	1	+	+	CCONJ
ma-117	82	2	b(xk)∆tk	b(xk)∆tk	ADJ
ma-117	82	3	+	+	CCONJ
ma-117	82	4	σ(xk)∆bk	σ(xk)∆bk	ADJ
ma-117	82	5	+	+	CCONJ
ma-117	82	6	1	1	NUM
ma-117	82	7	2	2	NUM
ma-117	82	8	σ(xk)σ′(xk)(∆bk	σ(xk)σ′(xk)(∆bk	NOUN
ma-117	82	9	−	−	PROPN
ma-117	82	10	∆tk	∆tk	PROPN
ma-117	82	11	)	)	PUNCT
ma-117	82	12	x(0	x(0	PROPN
ma-117	82	13	)	)	PUNCT
ma-117	82	14	=	=	SYM
ma-117	82	15	x0	x0	PROPN
ma-117	82	16	(	(	PUNCT
ma-117	82	17	2.4	2.4	NUM
ma-117	82	18	)	)	PUNCT
ma-117	82	19	remark	remark	NOUN
ma-117	82	20	2.1	2.1	NUM
ma-117	82	21	.	.	PUNCT
ma-117	83	1	it	it	PRON
ma-117	83	2	should	should	AUX
ma-117	83	3	be	be	AUX
ma-117	83	4	noted	note	VERB
ma-117	83	5	that	that	SCONJ
ma-117	83	6	the	the	DET
ma-117	83	7	euler	euler	PROPN
ma-117	83	8	-	-	PUNCT
ma-117	83	9	maruyama	maruyama	NOUN
ma-117	83	10	scheme	scheme	NOUN
ma-117	83	11	converges	converge	NOUN
ma-117	83	12	strongly	strongly	ADV
ma-117	83	13	up	up	ADP
ma-117	83	14	to	to	ADP
ma-117	83	15	the	the	DET
ma-117	83	16	order	order	NOUN
ma-117	83	17	1	1	NUM
ma-117	83	18	2	2	NUM
ma-117	83	19	while	while	SCONJ
ma-117	83	20	that	that	PRON
ma-117	83	21	of	of	ADP
ma-117	83	22	milshtein	milshtein	NOUN
ma-117	83	23	converges	converge	VERB
ma-117	83	24	up	up	ADP
ma-117	83	25	to	to	ADP
ma-117	83	26	the	the	DET
ma-117	83	27	order	order	NOUN
ma-117	83	28	1	1	NUM
ma-117	83	29	.	.	NOUN
ma-117	83	30	3	3	NUM
ma-117	83	31	.	.	NOUN
ma-117	83	32	numerical	numerical	ADJ
ma-117	83	33	stabilities	stability	NOUN
ma-117	83	34	of	of	ADP
ma-117	83	35	vasicek	vasicek	PROPN
ma-117	83	36	model	model	NOUN
ma-117	83	37	3.1	3.1	NUM
ma-117	83	38	.	.	PUNCT
ma-117	83	39	explicit	explicit	ADJ
ma-117	83	40	solution	solution	NOUN
ma-117	83	41	.	.	PUNCT
ma-117	84	1	the	the	DET
ma-117	84	2	vasicek	vasicek	PROPN
ma-117	84	3	model	model	NOUN
ma-117	84	4	(	(	PUNCT
ma-117	84	5	1977	1977	NUM
ma-117	84	6	)	)	PUNCT
ma-117	84	7	is	be	AUX
ma-117	84	8	one	one	NUM
ma-117	84	9	of	of	ADP
ma-117	84	10	the	the	DET
ma-117	84	11	first	first	ADJ
ma-117	84	12	stochastic	stochastic	ADJ
ma-117	84	13	interest	interest	NOUN
ma-117	84	14	rate	rate	NOUN
ma-117	84	15	models.it	models.it	PROPN
ma-117	84	16	is	be	AUX
ma-117	84	17	a	a	DET
ma-117	84	18	gaussian	gaussian	ADJ
ma-117	84	19	process	process	NOUN
ma-117	84	20	generalizing	generalize	VERB
ma-117	84	21	the	the	DET
ma-117	84	22	ornstein	ornstein	PROPN
ma-117	84	23	-	-	PUNCT
ma-117	84	24	unlenbeck	unlenbeck	PROPN
ma-117	84	25	model	model	NOUN
ma-117	84	26	and	and	CCONJ
ma-117	84	27	explains	explain	VERB
ma-117	84	28	the	the	DET
ma-117	84	29	observedempirical	observedempirical	ADJ
ma-117	84	30	mean	mean	ADJ
ma-117	84	31	reversion	reversion	NOUN
ma-117	84	32	effect	effect	NOUN
ma-117	84	33	on	on	ADP
ma-117	84	34	interest	interest	NOUN
ma-117	84	35	rate	rate	NOUN
ma-117	84	36	curves	curve	NOUN
ma-117	84	37	[	[	X
ma-117	84	38	15	15	NUM
ma-117	84	39	]	]	PUNCT
ma-117	84	40	,	,	PUNCT
ma-117	84	41	this	this	DET
ma-117	84	42	model	model	NOUN
ma-117	84	43	looks	look	VERB
ma-117	84	44	like:dxt	like:dxt	NOUN
ma-117	84	45	=	=	SYM
ma-117	84	46	(	(	PUNCT
ma-117	84	47	θ1	θ1	NOUN
ma-117	84	48	−	−	PROPN
ma-117	84	49	θ2xt)dt	θ2xt)dt	NOUN
ma-117	84	50	+	+	NUM
ma-117	84	51	θ3dbt	θ3dbt	PROPN
ma-117	84	52	x(0	x(0	NUM
ma-117	84	53	)	)	PUNCT
ma-117	85	1	=	=	PUNCT
ma-117	86	1	x0	x0	PROPN
ma-117	86	2	∀θ1	∀θ1	NOUN
ma-117	86	3	,	,	PUNCT
ma-117	86	4	θ2	θ2	PROPN
ma-117	86	5	et	et	NOUN
ma-117	86	6	θ3	θ3	PROPN
ma-117	86	7	>	>	X
ma-117	86	8	0	0	PUNCT
ma-117	87	1	(	(	PUNCT
ma-117	87	2	3.1	3.1	NUM
ma-117	87	3	)	)	PUNCT
ma-117	87	4	with	with	ADP
ma-117	87	5	xt	xt	PROPN
ma-117	87	6	:	:	PUNCT
ma-117	87	7	the	the	DET
ma-117	87	8	instant	instant	ADJ
ma-117	87	9	interest	interest	NOUN
ma-117	87	10	rate	rate	NOUN
ma-117	87	11	;	;	PUNCT
ma-117	87	12	θ2	θ2	PROPN
ma-117	87	13	:	:	PUNCT
ma-117	87	14	mean	mean	VERB
ma-117	87	15	reversion	reversion	NOUN
ma-117	87	16	rate	rate	NOUN
ma-117	87	17	;	;	PUNCT
ma-117	87	18	θ1	θ1	NOUN
ma-117	87	19	:	:	PUNCT
ma-117	87	20	the	the	DET
ma-117	87	21	long	long	ADJ
ma-117	87	22	-	-	PUNCT
ma-117	87	23	term	term	NOUN
ma-117	87	24	average	average	NOUN
ma-117	87	25	and	and	CCONJ
ma-117	87	26	θ3	θ3	NOUN
ma-117	87	27	:	:	PUNCT
ma-117	87	28	thevolatility.the	thevolatility.the	DET
ma-117	87	29	analytical	analytical	ADJ
ma-117	87	30	solution	solution	NOUN
ma-117	87	31	of	of	ADP
ma-117	87	32	(	(	PUNCT
ma-117	87	33	3.1	3.1	NUM
ma-117	87	34	)	)	PUNCT
ma-117	87	35	model	model	NOUN
ma-117	87	36	is	be	AUX
ma-117	87	37	:	:	PUNCT
ma-117	87	38	xt	xt	PROPN
ma-117	87	39	=	=	SYM
ma-117	87	40	θ1	θ1	PROPN
ma-117	87	41	θ2	θ2	PROPN
ma-117	87	42	+	+	CCONJ
ma-117	87	43	(	(	PUNCT
ma-117	87	44	x0	x0	PROPN
ma-117	87	45	−	−	PROPN
ma-117	87	46	θ1	θ1	PROPN
ma-117	87	47	θ2	θ2	PROPN
ma-117	87	48	)	)	PUNCT
ma-117	87	49	e−θ2	e−θ2	PROPN
ma-117	87	50	t	t	PROPN
ma-117	87	51	+	+	CCONJ
ma-117	87	52	θ3	θ3	PROPN
ma-117	87	53	∫	∫	PROPN
ma-117	88	1	+	+	PROPN
ma-117	88	2	∞	∞	PROPN
ma-117	88	3	0	0	NUM
ma-117	88	4	e−θ2(t−u)dbu	e−θ2(t−u)dbu	X
ma-117	88	5	(	(	PUNCT
ma-117	88	6	3.2	3.2	NUM
ma-117	88	7	)	)	PUNCT
ma-117	88	8	the	the	DET
ma-117	88	9	model	model	NOUN
ma-117	88	10	(	(	PUNCT
ma-117	88	11	3.1	3.1	NUM
ma-117	88	12	)	)	PUNCT
ma-117	88	13	is	be	AUX
ma-117	88	14	equivalent	equivalent	ADJ
ma-117	88	15	to	to	ADP
ma-117	88	16	the	the	PRON
ma-117	88	17	model:dxt	model:dxt	NOUN
ma-117	88	18	=	=	PUNCT
ma-117	88	19	θ(µ−xt)dt	θ(µ−xt)dt	NOUN
ma-117	88	20	+	+	PUNCT
ma-117	88	21	σdbt	σdbt	VERB
ma-117	88	22	x(0	x(0	PROPN
ma-117	88	23	)	)	PUNCT
ma-117	89	1	=	=	PUNCT
ma-117	89	2	x0	x0	PROPN
ma-117	89	3	(	(	PUNCT
ma-117	89	4	3.3	3.3	NUM
ma-117	89	5	)	)	PUNCT
ma-117	89	6	https://doi.org/10.28924/ada/ma.3.8	https://doi.org/10.28924/ada/ma.3.8	NUM
ma-117	89	7	eur	eur	NOUN
ma-117	89	8	.	.	PUNCT
ma-117	90	1	j.	j.	PROPN
ma-117	90	2	math	math	PROPN
ma-117	90	3	.	.	PUNCT
ma-117	91	1	anal	anal	PROPN
ma-117	91	2	.	.	PUNCT
ma-117	92	1	10.28924	10.28924	NUM
ma-117	92	2	/	/	SYM
ma-117	92	3	ada	ada	PROPN
ma-117	92	4	/	/	SYM
ma-117	92	5	ma.3.8	ma.3.8	PROPN
ma-117	92	6	5the	5the	DET
ma-117	92	7	solution	solution	NOUN
ma-117	92	8	of	of	ADP
ma-117	92	9	(	(	PUNCT
ma-117	92	10	3.3	3.3	NUM
ma-117	92	11	)	)	PUNCT
ma-117	92	12	has	have	AUX
ma-117	92	13	given	give	VERB
ma-117	92	14	by	by	ADP
ma-117	92	15	:	:	PUNCT
ma-117	92	16	xt	xt	PROPN
ma-117	92	17	=	=	SYM
ma-117	92	18	µ+	µ+	PUNCT
ma-117	92	19	(	(	PUNCT
ma-117	92	20	x0	x0	PROPN
ma-117	92	21	−	−	PROPN
ma-117	92	22	µ	µ	X
ma-117	92	23	)	)	PUNCT
ma-117	92	24	e−θt	e−θt	NOUN
ma-117	92	25	+	+	CCONJ
ma-117	92	26	θ	θ	PROPN
ma-117	92	27	∫	∫	PROPN
ma-117	93	1	t	t	PROPN
ma-117	93	2	0	0	NUM
ma-117	94	1	e−θ2(t−u)dbu	e−θ2(t−u)dbu	X
ma-117	94	2	(	(	PUNCT
ma-117	94	3	3.4	3.4	NUM
ma-117	94	4	)	)	PUNCT
ma-117	94	5	considering	consider	VERB
ma-117	94	6	the	the	DET
ma-117	94	7	solution	solution	NOUN
ma-117	94	8	of	of	ADP
ma-117	94	9	(	(	PUNCT
ma-117	94	10	3.2	3.2	NUM
ma-117	94	11	)	)	PUNCT
ma-117	94	12	,	,	PUNCT
ma-117	94	13	the	the	DET
ma-117	94	14	mean	mean	NOUN
ma-117	94	15	and	and	CCONJ
ma-117	94	16	the	the	DET
ma-117	94	17	mean	mean	ADJ
ma-117	94	18	-	-	PUNCT
ma-117	94	19	square	square	NOUN
ma-117	94	20	give	give	NOUN
ma-117	94	21	respectively	respectively	ADV
ma-117	94	22	:	:	PUNCT
ma-117	94	23	e[xt	e[xt	NOUN
ma-117	94	24	]	]	PUNCT
ma-117	94	25	=	=	SYM
ma-117	94	26	θ1	θ1	NOUN
ma-117	94	27	θ2	θ2	ADP
ma-117	94	28	∀	∀	NOUN
ma-117	94	29	θ2	θ2	ADV
ma-117	94	30	>	>	X
ma-117	94	31	0	0	PUNCT
ma-117	94	32	and	and	CCONJ
ma-117	94	33	v	v	NOUN
ma-117	94	34	(	(	PUNCT
ma-117	94	35	xt	xt	X
ma-117	94	36	)	)	PUNCT
ma-117	95	1	=	=	SYM
ma-117	95	2	θ2	θ2	ADP
ma-117	95	3	3	3	NUM
ma-117	95	4	2θ2	2θ2	NUM
ma-117	95	5	∀	∀	NOUN
ma-117	95	6	θ2	θ2	ADV
ma-117	95	7	>	>	X
ma-117	95	8	0	0	NUM
ma-117	95	9	which	which	PRON
ma-117	95	10	means	mean	VERB
ma-117	95	11	that	that	SCONJ
ma-117	95	12	the	the	DET
ma-117	95	13	stochastic	stochastic	ADJ
ma-117	95	14	process	process	NOUN
ma-117	95	15	xt	xt	PUNCT
ma-117	95	16	'	'	PUNCT
ma-117	95	17	n	n	CCONJ
ma-117	95	18	(	(	PUNCT
ma-117	95	19	θ1	θ1	NOUN
ma-117	95	20	θ2	θ2	PROPN
ma-117	95	21	,	,	PUNCT
ma-117	95	22	θ2	θ2	PROPN
ma-117	95	23	3	3	NUM
ma-117	95	24	2θ2	2θ2	NUM
ma-117	95	25	)	)	PUNCT
ma-117	95	26	by	by	ADP
ma-117	95	27	using	use	VERB
ma-117	95	28	some	some	DET
ma-117	95	29	properties	property	NOUN
ma-117	95	30	of	of	ADP
ma-117	95	31	brownian	brownian	ADJ
ma-117	95	32	motion	motion	NOUN
ma-117	95	33	,	,	PUNCT
ma-117	95	34	the	the	DET
ma-117	95	35	solution	solution	NOUN
ma-117	95	36	of	of	ADP
ma-117	95	37	the	the	DET
ma-117	95	38	model	model	NOUN
ma-117	95	39	(	(	PUNCT
ma-117	95	40	3.2	3.2	NUM
ma-117	95	41	)	)	PUNCT
ma-117	95	42	can	can	AUX
ma-117	95	43	be	be	AUX
ma-117	95	44	written	write	VERB
ma-117	95	45	asfollows	asfollow	VERB
ma-117	95	46	:	:	PUNCT
ma-117	95	47	xt	xt	PROPN
ma-117	95	48	=	=	SYM
ma-117	95	49	θ1	θ1	PROPN
ma-117	95	50	θ2	θ2	PROPN
ma-117	95	51	+	+	CCONJ
ma-117	95	52	θ3e	θ3e	NOUN
ma-117	95	53	−2θ2	−2θ2	NOUN
ma-117	95	54	t	t	NOUN
ma-117	95	55	√	√	NUM
ma-117	95	56	2θ2	2θ2	NUM
ma-117	95	57	b(e2θ2	b(e2θ2	PROPN
ma-117	95	58	t	t	PROPN
ma-117	95	59	)	)	PUNCT
ma-117	95	60	(	(	PUNCT
ma-117	95	61	3.5	3.5	NUM
ma-117	95	62	)	)	PUNCT
ma-117	95	63	now	now	ADV
ma-117	95	64	,	,	PUNCT
ma-117	95	65	we	we	PRON
ma-117	95	66	present	present	VERB
ma-117	95	67	some	some	DET
ma-117	95	68	numerical	numerical	ADJ
ma-117	95	69	stabilities	stability	NOUN
ma-117	95	70	conditions	condition	NOUN
ma-117	95	71	for	for	ADP
ma-117	95	72	the	the	DET
ma-117	95	73	system	system	NOUN
ma-117	95	74	(	(	PUNCT
ma-117	95	75	3.1	3.1	NUM
ma-117	95	76	)	)	PUNCT
ma-117	95	77	of	of	ADP
ma-117	95	78	some	some	DET
ma-117	95	79	numericalschemes	numericalscheme	NOUN
ma-117	95	80	(	(	PUNCT
ma-117	95	81	euler	euler	NOUN
ma-117	95	82	-	-	PUNCT
ma-117	95	83	maruyama	maruyama	NOUN
ma-117	95	84	,	,	PUNCT
ma-117	95	85	implicit	implicit	ADJ
ma-117	95	86	euler	euler	NOUN
ma-117	95	87	-	-	PUNCT
ma-117	95	88	maruyama	maruyama	NOUN
ma-117	95	89	and	and	CCONJ
ma-117	95	90	milshtein	milshtein	PROPN
ma-117	95	91	)	)	PUNCT
ma-117	95	92	and	and	CCONJ
ma-117	95	93	the	the	DET
ma-117	95	94	proofs	proof	NOUN
ma-117	95	95	of	of	ADP
ma-117	95	96	these	these	DET
ma-117	95	97	basedon	basedon	NOUN
ma-117	95	98	the	the	DET
ma-117	95	99	approach	approach	NOUN
ma-117	95	100	described	describe	VERB
ma-117	95	101	in	in	ADP
ma-117	95	102	[	[	X
ma-117	95	103	5	5	NUM
ma-117	95	104	]	]	PUNCT
ma-117	95	105	.	.	PUNCT
ma-117	96	1	3.2	3.2	NUM
ma-117	96	2	.	.	PUNCT
ma-117	96	3	euler	euler	NOUN
ma-117	96	4	-	-	PUNCT
ma-117	96	5	maruyama	maruyama	PROPN
ma-117	96	6	scheme	scheme	NOUN
ma-117	96	7	stabilities	stability	NOUN
ma-117	96	8	.	.	PUNCT
ma-117	97	1	the	the	DET
ma-117	97	2	euler	euler	NOUN
ma-117	97	3	-	-	PUNCT
ma-117	97	4	maruyama	maruyama	NOUN
ma-117	97	5	scheme	scheme	NOUN
ma-117	97	6	associated	associate	VERB
ma-117	97	7	to	to	ADP
ma-117	97	8	the	the	DET
ma-117	97	9	system(3.1	system(3.1	NOUN
ma-117	97	10	)	)	PUNCT
ma-117	97	11	is	be	AUX
ma-117	97	12	:	:	PUNCT
ma-117	97	13	xemk+1	xemk+1	PROPN
ma-117	97	14	=	=	SYM
ma-117	97	15	xk	xk	PROPN
ma-117	97	16	+	+	CCONJ
ma-117	97	17	(	(	PUNCT
ma-117	97	18	θ1	θ1	NOUN
ma-117	97	19	−	−	PROPN
ma-117	97	20	θ2xk	θ2xk	NOUN
ma-117	97	21	)	)	PUNCT
ma-117	97	22	∆t	∆t	PROPN
ma-117	98	1	+	+	CCONJ
ma-117	98	2	θ3∆bk	θ3∆bk	NOUN
ma-117	98	3	xemk+1	xemk+1	PUNCT
ma-117	98	4	=	=	PUNCT
ma-117	98	5	θ1∆t	θ1∆t	NOUN
ma-117	98	6	+	+	CCONJ
ma-117	98	7	(	(	PUNCT
ma-117	98	8	1−	1−	NUM
ma-117	98	9	θ2∆t)xk	θ2∆t)xk	NOUN
ma-117	98	10	+	+	CCONJ
ma-117	98	11	θ3	θ3	NOUN
ma-117	98	12	√	√	PROPN
ma-117	98	13	∆tzk	∆tzk	NOUN
ma-117	98	14	(	(	PUNCT
ma-117	98	15	3.6	3.6	NUM
ma-117	98	16	)	)	PUNCT
ma-117	98	17	3.2.1	3.2.1	NUM
ma-117	98	18	.	.	PUNCT
ma-117	99	1	mean	mean	VERB
ma-117	99	2	stability	stability	NOUN
ma-117	99	3	of	of	ADP
ma-117	99	4	euler	euler	NOUN
ma-117	99	5	-	-	PUNCT
ma-117	99	6	maruyama	maruyama	NOUN
ma-117	99	7	scheme	scheme	NOUN
ma-117	99	8	.	.	PUNCT
ma-117	100	1	theorem	theorem	VERB
ma-117	100	2	3.1	3.1	NUM
ma-117	100	3	.	.	PUNCT
ma-117	101	1	(	(	PUNCT
ma-117	101	2	mean	mean	VERB
ma-117	101	3	stability	stability	NOUN
ma-117	101	4	of	of	ADP
ma-117	101	5	euler	euler	NOUN
ma-117	101	6	-	-	PUNCT
ma-117	101	7	maruyama	maruyama	NOUN
ma-117	101	8	scheme	scheme	NOUN
ma-117	101	9	)	)	PUNCT
ma-117	101	10	the	the	DET
ma-117	101	11	euler	euler	NOUN
ma-117	101	12	-	-	PUNCT
ma-117	101	13	maruyama	maruyama	NOUN
ma-117	101	14	scheme	scheme	NOUN
ma-117	101	15	(	(	PUNCT
ma-117	101	16	3.6	3.6	NUM
ma-117	101	17	)	)	PUNCT
ma-117	101	18	of	of	ADP
ma-117	101	19	the	the	DET
ma-117	101	20	vasicek	vasicek	PROPN
ma-117	101	21	model	model	NOUN
ma-117	101	22	(	(	PUNCT
ma-117	101	23	3.1	3.1	NUM
ma-117	101	24	)	)	PUNCT
ma-117	101	25	is	be	AUX
ma-117	101	26	mean	mean	VERB
ma-117	101	27	asymptotically	asymptotically	ADV
ma-117	101	28	stable	stable	ADJ
ma-117	101	29	if	if	SCONJ
ma-117	101	30	:	:	PUNCT
ma-117	101	31	e	e	X
ma-117	101	32	[	[	PUNCT
ma-117	101	33	xemk+1	xemk+1	X
ma-117	101	34	]	]	PUNCT
ma-117	102	1	=	=	SYM
ma-117	102	2	(	(	PUNCT
ma-117	102	3	1−	1−	NUM
ma-117	102	4	θ2∆t)k+1e[x0	θ2∆t)k+1e[x0	NOUN
ma-117	102	5	]	]	PUNCT
ma-117	102	6	+	+	CCONJ
ma-117	102	7	θ1∆t	θ1∆t	NOUN
ma-117	102	8	[	[	PUNCT
ma-117	102	9	k+1∑	k+1∑	PROPN
ma-117	102	10	i=0	i=0	PROPN
ma-117	102	11	(	(	PUNCT
ma-117	102	12	1−	1−	NUM
ma-117	102	13	θ2∆t)i	θ2∆t)i	X
ma-117	102	14	]	]	PUNCT
ma-117	102	15	(	(	PUNCT
ma-117	102	16	3.7	3.7	NUM
ma-117	102	17	)	)	PUNCT
ma-117	102	18	with	with	AUX
ma-117	102	19	|1−	|1−	VERB
ma-117	102	20	θ2∆t|	θ2∆t|	PUNCT
ma-117	102	21	<	<	X
ma-117	102	22	1	1	NUM
ma-117	102	23	and	and	CCONJ
ma-117	102	24	lim	lim	PROPN
ma-117	102	25	∆t→0	∆t→0	PROPN
ma-117	102	26	(	(	PUNCT
ma-117	102	27	lim	lim	PROPN
ma-117	102	28	k→+∞	k→+∞	PROPN
ma-117	102	29	e	e	PROPN
ma-117	102	30	[	[	PUNCT
ma-117	102	31	xemk+1	xemk+1	X
ma-117	102	32	]	]	X
ma-117	102	33	)	)	PUNCT
ma-117	102	34	=	=	SYM
ma-117	102	35	θ1	θ1	NOUN
ma-117	102	36	θ2	θ2	PROPN
ma-117	102	37	https://doi.org/10.28924/ada/ma.3.8	https://doi.org/10.28924/ada/ma.3.8	PROPN
ma-117	102	38	eur	eur	PROPN
ma-117	102	39	.	.	PUNCT
ma-117	103	1	j.	j.	PROPN
ma-117	103	2	math	math	PROPN
ma-117	103	3	.	.	PUNCT
ma-117	104	1	anal	anal	PROPN
ma-117	104	2	.	.	PUNCT
ma-117	105	1	10.28924	10.28924	NUM
ma-117	105	2	/	/	SYM
ma-117	105	3	ada	ada	PROPN
ma-117	105	4	/	/	SYM
ma-117	105	5	ma.3.8	ma.3.8	PROPN
ma-117	105	6	6	6	NUM
ma-117	105	7	proof	proof	NOUN
ma-117	105	8	.	.	PUNCT
ma-117	106	1	to	to	PART
ma-117	106	2	prove	prove	VERB
ma-117	106	3	the	the	DET
ma-117	106	4	theorem	theorem	NOUN
ma-117	106	5	,	,	PUNCT
ma-117	106	6	we	we	PRON
ma-117	106	7	start	start	VERB
ma-117	106	8	by	by	ADP
ma-117	106	9	evaluating	evaluate	VERB
ma-117	106	10	the	the	DET
ma-117	106	11	mean	mean	NOUN
ma-117	106	12	of	of	ADP
ma-117	106	13	the	the	DET
ma-117	106	14	(	(	PUNCT
ma-117	106	15	3.1	3.1	NUM
ma-117	106	16	)	)	PUNCT
ma-117	106	17	equation	equation	NOUN
ma-117	106	18	using	use	VERB
ma-117	106	19	theapproach	theapproach	NOUN
ma-117	106	20	defined	define	VERB
ma-117	106	21	in	in	ADP
ma-117	106	22	[	[	X
ma-117	106	23	5	5	NUM
ma-117	106	24	]	]	PUNCT
ma-117	106	25	.	.	PUNCT
ma-117	107	1	in	in	ADP
ma-117	107	2	effect	effect	NOUN
ma-117	107	3	,	,	PUNCT
ma-117	107	4	e	e	X
ma-117	107	5	[	[	PUNCT
ma-117	107	6	xemk+1	xemk+1	X
ma-117	107	7	]	]	PUNCT
ma-117	107	8	=	=	SYM
ma-117	107	9	e	e	X
ma-117	107	10	[	[	PUNCT
ma-117	107	11	θ1∆t	θ1∆t	NOUN
ma-117	107	12	+	+	ADJ
ma-117	107	13	xk	xk	X
ma-117	107	14	(	(	PUNCT
ma-117	107	15	1−	1−	NUM
ma-117	107	16	θ2∆t	θ2∆t	NOUN
ma-117	107	17	)	)	PUNCT
ma-117	107	18	+	+	CCONJ
ma-117	107	19	θ3	θ3	NOUN
ma-117	107	20	√	√	PROPN
ma-117	107	21	∆tzk	∆tzk	NOUN
ma-117	107	22	]	]	PUNCT
ma-117	108	1	=	=	PUNCT
ma-117	108	2	e	e	X
ma-117	109	1	[	[	X
ma-117	109	2	θ1∆t	θ1∆t	X
ma-117	109	3	]	]	PUNCT
ma-117	109	4	+	+	CCONJ
ma-117	109	5	e	e	X
ma-117	109	6	[	[	X
ma-117	109	7	xk	xk	X
ma-117	109	8	(	(	PUNCT
ma-117	109	9	1−	1−	NUM
ma-117	109	10	θ2∆t	θ2∆t	NOUN
ma-117	109	11	)	)	PUNCT
ma-117	109	12	]	]	PUNCT
ma-117	110	1	+	+	CCONJ
ma-117	110	2	e	e	X
ma-117	110	3	[	[	PUNCT
ma-117	110	4	θ3	θ3	NOUN
ma-117	110	5	√	√	PROPN
ma-117	110	6	∆tzk	∆tzk	NOUN
ma-117	110	7	]	]	PUNCT
ma-117	111	1	=	=	PUNCT
ma-117	111	2	e	e	X
ma-117	112	1	[	[	X
ma-117	112	2	θ1∆t	θ1∆t	X
ma-117	112	3	]	]	X
ma-117	112	4	+	+	CCONJ
ma-117	112	5	(	(	PUNCT
ma-117	112	6	1−	1−	NUM
ma-117	112	7	θ2∆t)e	θ2∆t)e	NOUN
ma-117	113	1	[	[	X
ma-117	113	2	xk	xk	X
ma-117	113	3	]	]	X
ma-117	113	4	+	+	PUNCT
ma-117	113	5	0	0	X
ma-117	113	6	]	]	PUNCT
ma-117	113	7	with	with	ADP
ma-117	113	8	zk	zk	PROPN
ma-117	113	9	'	'	PART
ma-117	113	10	n	n	CCONJ
ma-117	113	11	(	(	PUNCT
ma-117	113	12	0	0	NUM
ma-117	113	13	,	,	PUNCT
ma-117	113	14	1	1	NUM
ma-117	113	15	)	)	PUNCT
ma-117	113	16	=	=	PUNCT
ma-117	113	17	θ1∆t	θ1∆t	NOUN
ma-117	113	18	+	+	CCONJ
ma-117	113	19	(	(	PUNCT
ma-117	113	20	1−	1−	NUM
ma-117	113	21	θ2∆t)e[xk	θ2∆t)e[xk	ADP
ma-117	113	22	]	]	PUNCT
ma-117	114	1	=	=	PUNCT
ma-117	114	2	θ1∆t	θ1∆t	NOUN
ma-117	114	3	+	+	CCONJ
ma-117	114	4	(	(	PUNCT
ma-117	114	5	1−	1−	NUM
ma-117	114	6	θ2∆t	θ2∆t	NOUN
ma-117	114	7	)	)	PUNCT
ma-117	114	8	{	{	PUNCT
ma-117	114	9	(	(	PUNCT
ma-117	114	10	1−	1−	NUM
ma-117	114	11	θ2∆t)e[xk−1	θ2∆t)e[xk−1	X
ma-117	114	12	]	]	PUNCT
ma-117	115	1	+	+	CCONJ
ma-117	115	2	θ1∆t	θ1∆t	NOUN
ma-117	115	3	}	}	PUNCT
ma-117	115	4	=	=	PUNCT
ma-117	115	5	θ1∆t	θ1∆t	NOUN
ma-117	115	6	+	+	SYM
ma-117	115	7	θ1∆t(1−	θ1∆t(1−	X
ma-117	115	8	θ2∆t	θ2∆t	NOUN
ma-117	115	9	)	)	PUNCT
ma-117	116	1	+	+	CCONJ
ma-117	116	2	(	(	PUNCT
ma-117	116	3	1−	1−	NUM
ma-117	116	4	θ2∆t)2e[xk−1	θ2∆t)2e[xk−1	NOUN
ma-117	116	5	]	]	X
ma-117	116	6	=	=	PUNCT
ma-117	116	7	θ1∆t(1	θ1∆t(1	PROPN
ma-117	116	8	+	+	CCONJ
ma-117	116	9	(	(	PUNCT
ma-117	116	10	1−	1−	NUM
ma-117	116	11	θ2∆t	θ2∆t	NOUN
ma-117	116	12	)	)	PUNCT
ma-117	116	13	)	)	PUNCT
ma-117	117	1	+	+	CCONJ
ma-117	117	2	(	(	PUNCT
ma-117	117	3	1−	1−	NUM
ma-117	117	4	θ2∆t)2e[xk−1	θ2∆t)2e[xk−1	NOUN
ma-117	117	5	]	]	X
ma-117	117	6	=	=	PUNCT
ma-117	117	7	θ1∆t(1	θ1∆t(1	PROPN
ma-117	117	8	+	+	CCONJ
ma-117	117	9	(	(	PUNCT
ma-117	117	10	1−	1−	NUM
ma-117	117	11	θ2∆t	θ2∆t	NOUN
ma-117	117	12	)	)	PUNCT
ma-117	117	13	)	)	PUNCT
ma-117	118	1	+	+	CCONJ
ma-117	118	2	(	(	PUNCT
ma-117	118	3	1−	1−	NUM
ma-117	118	4	θ2∆t)2	θ2∆t)2	X
ma-117	118	5	{	{	PUNCT
ma-117	118	6	(	(	PUNCT
ma-117	118	7	1−	1−	NUM
ma-117	118	8	θ2∆t)e[xk−2	θ2∆t)e[xk−2	NOUN
ma-117	118	9	]	]	X
ma-117	118	10	+	+	PUNCT
ma-117	118	11	θ1∆t	θ1∆t	NOUN
ma-117	118	12	}	}	PUNCT
ma-117	118	13	=	=	SYM
ma-117	118	14	θ1∆t(1	θ1∆t(1	PROPN
ma-117	118	15	+	+	CCONJ
ma-117	118	16	(	(	PUNCT
ma-117	118	17	1−	1−	NUM
ma-117	118	18	θ2∆t	θ2∆t	NOUN
ma-117	118	19	)	)	PUNCT
ma-117	118	20	)	)	PUNCT
ma-117	119	1	+	+	CCONJ
ma-117	120	1	θ1∆(1−	θ1∆(1−	PROPN
ma-117	120	2	θ2∆t)2	θ2∆t)2	X
ma-117	120	3	+	+	CCONJ
ma-117	120	4	(	(	PUNCT
ma-117	120	5	1−	1−	NUM
ma-117	120	6	θ2∆t)3e[xk−2	θ2∆t)3e[xk−2	PROPN
ma-117	120	7	]	]	X
ma-117	120	8	=	=	PUNCT
ma-117	120	9	θ1∆t(1	θ1∆t(1	PROPN
ma-117	120	10	+	+	CCONJ
ma-117	120	11	(	(	PUNCT
ma-117	120	12	1−	1−	NUM
ma-117	120	13	θ2∆t	θ2∆t	NOUN
ma-117	120	14	)	)	PUNCT
ma-117	121	1	+	+	CCONJ
ma-117	121	2	(	(	PUNCT
ma-117	121	3	1−	1−	NUM
ma-117	121	4	θ2∆t)2	θ2∆t)2	NOUN
ma-117	121	5	)	)	PUNCT
ma-117	122	1	+	+	CCONJ
ma-117	122	2	(	(	PUNCT
ma-117	122	3	1−	1−	NUM
ma-117	122	4	θ2∆t)3e[xk−2	θ2∆t)3e[xk−2	PROPN
ma-117	122	5	]	]	X
ma-117	122	6	=	=	PUNCT
ma-117	122	7	θ1∆t(1	θ1∆t(1	PROPN
ma-117	122	8	+	+	CCONJ
ma-117	122	9	(	(	PUNCT
ma-117	122	10	1−	1−	NUM
ma-117	122	11	θ2∆t	θ2∆t	NOUN
ma-117	122	12	)	)	PUNCT
ma-117	123	1	+	+	CCONJ
ma-117	123	2	(	(	PUNCT
ma-117	123	3	1−	1−	NUM
ma-117	123	4	θ2∆t)2	θ2∆t)2	X
ma-117	124	1	+	+	CCONJ
ma-117	124	2	·	·	PUNCT
ma-117	124	3	·	·	PUNCT
ma-117	124	4	·	·	PUNCT
ma-117	124	5	+	+	PUNCT
ma-117	124	6	(	(	PUNCT
ma-117	124	7	1−	1−	NUM
ma-117	124	8	θ2∆t))k+1	θ2∆t))k+1	NUM
ma-117	124	9	+	+	CCONJ
ma-117	124	10	(	(	PUNCT
ma-117	124	11	1−	1−	NUM
ma-117	124	12	θ2∆t)k+1e[x0	θ2∆t)k+1e[x0	NOUN
ma-117	124	13	]	]	X
ma-117	124	14	=	=	SYM
ma-117	124	15	(	(	PUNCT
ma-117	124	16	1−	1−	NUM
ma-117	124	17	θ2∆t)k+1e[x0	θ2∆t)k+1e[x0	NOUN
ma-117	124	18	]	]	PUNCT
ma-117	124	19	+	+	CCONJ
ma-117	124	20	θ1∆t	θ1∆t	NOUN
ma-117	124	21	[	[	PUNCT
ma-117	124	22	k+1∑	k+1∑	PROPN
ma-117	124	23	i=0	i=0	PROPN
ma-117	124	24	(	(	PUNCT
ma-117	124	25	1−	1−	NUM
ma-117	124	26	θ2∆t)i	θ2∆t)i	PRON
ma-117	124	27	]	]	PUNCT
ma-117	124	28	using	use	VERB
ma-117	124	29	the	the	DET
ma-117	124	30	theory	theory	NOUN
ma-117	124	31	of	of	ADP
ma-117	124	32	geometric	geometric	ADJ
ma-117	124	33	sequences	sequence	NOUN
ma-117	124	34	and	and	CCONJ
ma-117	124	35	series	series	NOUN
ma-117	124	36	,	,	PUNCT
ma-117	124	37	we	we	PRON
ma-117	124	38	get	get	VERB
ma-117	124	39	:	:	PUNCT
ma-117	124	40	e	e	X
ma-117	124	41	[	[	PUNCT
ma-117	124	42	xemk+1	xemk+1	X
ma-117	124	43	]	]	PUNCT
ma-117	124	44	=	=	SYM
ma-117	124	45	(	(	PUNCT
ma-117	124	46	1−	1−	NUM
ma-117	124	47	θ2∆t)k+1e[x0	θ2∆t)k+1e[x0	NOUN
ma-117	124	48	]	]	PUNCT
ma-117	125	1	+	+	CCONJ
ma-117	125	2	θ1∆t	θ1∆t	NOUN
ma-117	125	3	(	(	PUNCT
ma-117	125	4	(	(	PUNCT
ma-117	125	5	1−	1−	NUM
ma-117	125	6	(	(	PUNCT
ma-117	125	7	1−	1−	NUM
ma-117	125	8	θ2∆t)k+1	θ2∆t)k+1	PROPN
ma-117	125	9	)	)	PUNCT
ma-117	125	10	1−	1−	NUM
ma-117	125	11	(	(	PUNCT
ma-117	125	12	1−	1−	NUM
ma-117	125	13	θ2∆t	θ2∆t	NOUN
ma-117	125	14	)	)	PUNCT
ma-117	125	15	)	)	PUNCT
ma-117	126	1	(	(	PUNCT
ma-117	126	2	3.8	3.8	NUM
ma-117	126	3	)	)	PUNCT
ma-117	126	4	as	as	ADP
ma-117	126	5	the	the	DET
ma-117	126	6	identity	identity	NOUN
ma-117	126	7	(	(	PUNCT
ma-117	126	8	3.8	3.8	NUM
ma-117	126	9	)	)	PUNCT
ma-117	126	10	represents	represent	VERB
ma-117	126	11	a	a	DET
ma-117	126	12	geometric	geometric	ADJ
ma-117	126	13	sequence	sequence	NOUN
ma-117	126	14	,	,	PUNCT
ma-117	126	15	we	we	PRON
ma-117	126	16	have	have	VERB
ma-117	126	17	that	that	SCONJ
ma-117	126	18	it	it	PRON
ma-117	126	19	converges	converge	VERB
ma-117	126	20	if	if	SCONJ
ma-117	126	21	|1−	|1−	VERB
ma-117	126	22	θ2∆t|	θ2∆t|	PRON
ma-117	126	23	<	<	X
ma-117	126	24	1	1	NUM
ma-117	126	25	by	by	ADP
ma-117	126	26	calculating	calculate	VERB
ma-117	126	27	the	the	DET
ma-117	126	28	limit	limit	NOUN
ma-117	126	29	of	of	ADP
ma-117	126	30	the	the	DET
ma-117	126	31	(	(	PUNCT
ma-117	126	32	3.8	3.8	NUM
ma-117	126	33	)	)	PUNCT
ma-117	126	34	,	,	PUNCT
ma-117	126	35	for	for	ADP
ma-117	126	36	∆t	∆t	PROPN
ma-117	126	37	→	→	SYM
ma-117	126	38	0	0	NUM
ma-117	126	39	and	and	CCONJ
ma-117	126	40	k	k	PROPN
ma-117	126	41	→	→	SYM
ma-117	126	42	+	+	NOUN
ma-117	126	43	∞	∞	PROPN
ma-117	126	44	,	,	PUNCT
ma-117	126	45	we	we	PRON
ma-117	126	46	get	get	VERB
ma-117	126	47	:	:	PUNCT
ma-117	126	48	lim	lim	PROPN
ma-117	126	49	∆t→0	∆t→0	PROPN
ma-117	126	50	(	(	PUNCT
ma-117	126	51	lim	lim	PROPN
ma-117	126	52	k→+∞	k→+∞	PROPN
ma-117	126	53	e	e	PROPN
ma-117	126	54	[	[	PUNCT
ma-117	126	55	xemk+1	xemk+1	X
ma-117	126	56	]	]	X
ma-117	126	57	)	)	PUNCT
ma-117	126	58	=	=	SYM
ma-117	126	59	θ1	θ1	PROPN
ma-117	126	60	θ2	θ2	PROPN
ma-117	126	61	�	�	PROPN
ma-117	126	62	3.2.2	3.2.2	NUM
ma-117	126	63	.	.	PUNCT
ma-117	127	1	mean	mean	ADJ
ma-117	127	2	-	-	PUNCT
ma-117	127	3	square	square	ADJ
ma-117	127	4	stability	stability	NOUN
ma-117	127	5	of	of	ADP
ma-117	127	6	euler	euler	NOUN
ma-117	127	7	-	-	PUNCT
ma-117	127	8	maruyama	maruyama	NOUN
ma-117	127	9	scheme	scheme	NOUN
ma-117	127	10	.	.	PUNCT
ma-117	128	1	theorem	theorem	VERB
ma-117	128	2	3.2	3.2	NUM
ma-117	128	3	.	.	PUNCT
ma-117	129	1	(	(	PUNCT
ma-117	129	2	mean	mean	ADJ
ma-117	129	3	-	-	PUNCT
ma-117	129	4	square	square	ADJ
ma-117	129	5	stability	stability	NOUN
ma-117	129	6	of	of	ADP
ma-117	129	7	euler	euler	NOUN
ma-117	129	8	-	-	PUNCT
ma-117	129	9	maruyama	maruyama	NOUN
ma-117	129	10	scheme	scheme	NOUN
ma-117	129	11	)	)	PUNCT
ma-117	129	12	the	the	DET
ma-117	129	13	euler	euler	NOUN
ma-117	129	14	-	-	PUNCT
ma-117	129	15	maruyama	maruyama	NOUN
ma-117	129	16	scheme(3.6	scheme(3.6	NOUN
ma-117	129	17	)	)	PUNCT
ma-117	129	18	of	of	ADP
ma-117	129	19	the	the	DET
ma-117	129	20	vasicek	vasicek	PROPN
ma-117	129	21	model	model	NOUN
ma-117	129	22	(	(	PUNCT
ma-117	129	23	3.1	3.1	NUM
ma-117	129	24	)	)	PUNCT
ma-117	129	25	is	be	AUX
ma-117	129	26	mean	mean	ADJ
ma-117	129	27	-	-	PUNCT
ma-117	129	28	square	square	ADJ
ma-117	129	29	asymptotically	asymptotically	ADV
ma-117	129	30	stable	stable	ADJ
ma-117	129	31	if	if	SCONJ
ma-117	129	32	:	:	PUNCT
ma-117	129	33	e	e	X
ma-117	130	1	[	[	AUX
ma-117	130	2	∣∣xemk+1	∣∣xemk+1	VERB
ma-117	130	3	∣∣2	∣∣2	NOUN
ma-117	130	4	]	]	X
ma-117	130	5	=	=	PUNCT
ma-117	130	6	(	(	PUNCT
ma-117	130	7	1−	1−	NUM
ma-117	130	8	θ2∆t)2(k+1	θ2∆t)2(k+1	NOUN
ma-117	130	9	)	)	PUNCT
ma-117	130	10	e	e	NOUN
ma-117	130	11	(	(	PUNCT
ma-117	130	12	|x0|2	|x0|2	PROPN
ma-117	130	13	)	)	PUNCT
ma-117	131	1	+	+	CCONJ
ma-117	131	2	(	(	PUNCT
ma-117	131	3	θ2	θ2	ADV
ma-117	131	4	3	3	NUM
ma-117	131	5	+	+	CCONJ
ma-117	131	6	θ2	θ2	PROPN
ma-117	131	7	1∆t	1∆t	NUM
ma-117	131	8	)	)	PUNCT
ma-117	131	9	∆t	∆t	PROPN
ma-117	131	10	k+1∑	k+1∑	PROPN
ma-117	131	11	i=0	i=0	PROPN
ma-117	131	12	(	(	PUNCT
ma-117	131	13	1−	1−	NUM
ma-117	131	14	θ2∆t)2i	θ2∆t)2i	NUM
ma-117	131	15	https://doi.org/10.28924/ada/ma.3.8	https://doi.org/10.28924/ada/ma.3.8	VERB
ma-117	131	16	eur	eur	NOUN
ma-117	131	17	.	.	PUNCT
ma-117	132	1	j.	j.	PROPN
ma-117	132	2	math	math	PROPN
ma-117	132	3	.	.	PUNCT
ma-117	133	1	anal	anal	PROPN
ma-117	133	2	.	.	PUNCT
ma-117	134	1	10.28924	10.28924	NUM
ma-117	134	2	/	/	SYM
ma-117	134	3	ada	ada	PROPN
ma-117	134	4	/	/	SYM
ma-117	134	5	ma.3.8	ma.3.8	PROPN
ma-117	134	6	7	7	NUM
ma-117	134	7	and	and	CCONJ
ma-117	134	8	that	that	SCONJ
ma-117	134	9	the	the	DET
ma-117	134	10	following	follow	VERB
ma-117	134	11	two	two	NUM
ma-117	134	12	conditions	condition	NOUN
ma-117	134	13	are	be	AUX
ma-117	134	14	satisfied	satisfied	ADJ
ma-117	134	15	simultaneously	simultaneously	ADV
ma-117	134	16	:	:	PUNCT
ma-117	134	17	(	(	PUNCT
ma-117	134	18	1	1	X
ma-117	134	19	)	)	PUNCT
ma-117	134	20	|1−	|1−	VERB
ma-117	135	1	θ2∆t|	θ2∆t|	ADV
ma-117	135	2	<	<	X
ma-117	135	3	1	1	NUM
ma-117	135	4	(	(	PUNCT
ma-117	135	5	2	2	NUM
ma-117	135	6	)	)	PUNCT
ma-117	135	7	lim	lim	NOUN
ma-117	135	8	∆t→0	∆t→0	PROPN
ma-117	135	9	(	(	PUNCT
ma-117	135	10	lim	lim	PROPN
ma-117	135	11	k→+∞	k→+∞	PROPN
ma-117	135	12	e	e	PROPN
ma-117	135	13	[	[	PUNCT
ma-117	135	14	xemk+1	xemk+1	X
ma-117	135	15	]	]	X
ma-117	135	16	2	2	X
ma-117	135	17	)	)	PUNCT
ma-117	135	18	=	=	SYM
ma-117	135	19	θ2	θ2	ADP
ma-117	135	20	3	3	NUM
ma-117	135	21	2θ2	2θ2	NUM
ma-117	135	22	proof	proof	NOUN
ma-117	135	23	.	.	PUNCT
ma-117	136	1	as	as	ADP
ma-117	136	2	in	in	ADP
ma-117	136	3	the	the	DET
ma-117	136	4	previous	previous	ADJ
ma-117	136	5	theorem	theorem	NOUN
ma-117	136	6	,	,	PUNCT
ma-117	136	7	we	we	PRON
ma-117	136	8	start	start	VERB
ma-117	136	9	by	by	ADP
ma-117	136	10	calculating	calculate	VERB
ma-117	136	11	the	the	DET
ma-117	136	12	expression	expression	NOUN
ma-117	136	13	:	:	PUNCT
ma-117	137	1	e	e	X
ma-117	137	2	[	[	AUX
ma-117	137	3	∣∣xemk+1	∣∣xemk+1	VERB
ma-117	137	4	∣∣2	∣∣2	NUM
ma-117	137	5	]	]	PUNCT
ma-117	137	6	of	of	ADP
ma-117	137	7	vasicek	vasicek	PROPN
ma-117	137	8	model	model	NOUN
ma-117	137	9	of	of	ADP
ma-117	137	10	the	the	DET
ma-117	137	11	equation	equation	NOUN
ma-117	137	12	(	(	PUNCT
ma-117	137	13	3.1	3.1	NUM
ma-117	137	14	)	)	PUNCT
ma-117	137	15	.	.	PUNCT
ma-117	138	1	in	in	ADP
ma-117	138	2	effect	effect	NOUN
ma-117	138	3	,	,	PUNCT
ma-117	138	4	e	e	X
ma-117	138	5	[	[	AUX
ma-117	138	6	∣∣xemk+1	∣∣xemk+1	VERB
ma-117	138	7	∣∣2	∣∣2	NOUN
ma-117	138	8	]	]	X
ma-117	138	9	=	=	SYM
ma-117	138	10	|θ1∆t|2	|θ1∆t|2	X
ma-117	139	1	+	+	CCONJ
ma-117	139	2	e	e	X
ma-117	139	3	(	(	PUNCT
ma-117	139	4	|xk(1−	|xk(1−	PROPN
ma-117	139	5	θ2∆t)|)2	θ2∆t)|)2	PROPN
ma-117	139	6	+	+	PROPN
ma-117	139	7	θ2	θ2	PROPN
ma-117	139	8	3∆t	3∆t	NUM
ma-117	139	9	zk	zk	PROPN
ma-117	139	10	'	'	PUNCT
ma-117	139	11	n(0	n(0	PROPN
ma-117	139	12	,	,	PUNCT
ma-117	139	13	1	1	NUM
ma-117	139	14	)	)	PUNCT
ma-117	139	15	=	=	SYM
ma-117	139	16	(	(	PUNCT
ma-117	139	17	1−	1−	NUM
ma-117	139	18	θ2∆t)2e	θ2∆t)2e	NOUN
ma-117	139	19	(	(	PUNCT
ma-117	139	20	|xk	|xk	X
ma-117	139	21	|2	|2	NUM
ma-117	139	22	)	)	PUNCT
ma-117	139	23	+	+	CCONJ
ma-117	139	24	(	(	PUNCT
ma-117	139	25	θ2	θ2	ADV
ma-117	139	26	1∆t	1∆t	NUM
ma-117	139	27	+	+	CCONJ
ma-117	139	28	θ2	θ2	PROPN
ma-117	139	29	3)∆t	3)∆t	NUM
ma-117	139	30	=	=	SYM
ma-117	139	31	(	(	PUNCT
ma-117	139	32	1−	1−	NUM
ma-117	139	33	θ2∆t)2e	θ2∆t)2e	NOUN
ma-117	139	34	(	(	PUNCT
ma-117	139	35	|xk	|xk	X
ma-117	139	36	|2	|2	NUM
ma-117	139	37	)	)	PUNCT
ma-117	139	38	{	{	PUNCT
ma-117	139	39	e	e	X
ma-117	139	40	(	(	PUNCT
ma-117	139	41	|xk+1|2	|xk+1|2	PROPN
ma-117	139	42	)	)	PUNCT
ma-117	139	43	(	(	PUNCT
ma-117	139	44	1−	1−	NUM
ma-117	139	45	θ2∆t)2	θ2∆t)2	X
ma-117	140	1	+	+	CCONJ
ma-117	140	2	(	(	PUNCT
ma-117	140	3	θ2	θ2	ADV
ma-117	140	4	3	3	NUM
ma-117	140	5	+	+	CCONJ
ma-117	140	6	θ2	θ2	PROPN
ma-117	140	7	1∆t)∆t	1∆t)∆t	NUM
ma-117	140	8	}	}	PUNCT
ma-117	141	1	+	+	CCONJ
ma-117	141	2	(	(	PUNCT
ma-117	141	3	θ2	θ2	ADV
ma-117	141	4	3	3	NUM
ma-117	141	5	+	+	CCONJ
ma-117	141	6	θ2	θ2	PROPN
ma-117	141	7	1∆t)∆t	1∆t)∆t	NUM
ma-117	141	8	=	=	SYM
ma-117	141	9	(	(	PUNCT
ma-117	141	10	1−	1−	NUM
ma-117	141	11	θ2∆t)4e	θ2∆t)4e	ADV
ma-117	141	12	(	(	PUNCT
ma-117	141	13	|xk−1|2	|xk−1|2	X
ma-117	141	14	)	)	PUNCT
ma-117	142	1	+	+	CCONJ
ma-117	142	2	(	(	PUNCT
ma-117	142	3	θ2	θ2	ADV
ma-117	142	4	3	3	NUM
ma-117	142	5	+	+	CCONJ
ma-117	142	6	θ2	θ2	PROPN
ma-117	142	7	1∆t)∆t	1∆t)∆t	NUM
ma-117	142	8	[	[	PUNCT
ma-117	142	9	(	(	PUNCT
ma-117	142	10	1−	1−	NUM
ma-117	142	11	θ2∆t)2	θ2∆t)2	X
ma-117	143	1	+	+	CCONJ
ma-117	143	2	1	1	NUM
ma-117	143	3	]	]	PUNCT
ma-117	143	4	=	=	SYM
ma-117	143	5	(	(	PUNCT
ma-117	143	6	1−	1−	NUM
ma-117	143	7	θ2∆t)6e	θ2∆t)6e	NUM
ma-117	143	8	(	(	PUNCT
ma-117	143	9	|xk−2|2	|xk−2|2	PROPN
ma-117	143	10	)	)	PUNCT
ma-117	144	1	+	+	CCONJ
ma-117	144	2	(	(	PUNCT
ma-117	144	3	θ2	θ2	ADV
ma-117	144	4	3	3	NUM
ma-117	144	5	+	+	CCONJ
ma-117	144	6	θ2	θ2	PROPN
ma-117	144	7	1∆t)∆t[(1−	1∆t)∆t[(1−	NOUN
ma-117	144	8	θ2∆t)4	θ2∆t)4	ADP
ma-117	144	9	+	+	CCONJ
ma-117	144	10	(	(	PUNCT
ma-117	144	11	1−	1−	NUM
ma-117	144	12	θ2∆t)2	θ2∆t)2	X
ma-117	145	1	+	+	CCONJ
ma-117	145	2	1	1	NUM
ma-117	145	3	]	]	X
ma-117	145	4	=	=	SYM
ma-117	145	5	(	(	PUNCT
ma-117	145	6	1−	1−	NUM
ma-117	145	7	θ2∆t)8e	θ2∆t)8e	PROPN
ma-117	145	8	(	(	PUNCT
ma-117	145	9	|xk−3|2	|xk−3|2	NOUN
ma-117	145	10	)	)	PUNCT
ma-117	146	1	+	+	CCONJ
ma-117	146	2	(	(	PUNCT
ma-117	146	3	θ2	θ2	ADV
ma-117	146	4	3	3	NUM
ma-117	146	5	+	+	CCONJ
ma-117	146	6	θ2	θ2	PROPN
ma-117	146	7	1∆t)∆t	1∆t)∆t	NUM
ma-117	146	8	[	[	PUNCT
ma-117	146	9	(	(	PUNCT
ma-117	146	10	1−	1−	NUM
ma-117	146	11	θ2∆t)6	θ2∆t)6	NOUN
ma-117	146	12	+	+	CCONJ
ma-117	146	13	(	(	PUNCT
ma-117	146	14	1−	1−	NUM
ma-117	146	15	θ2∆t)4	θ2∆t)4	ADP
ma-117	146	16	+	+	CCONJ
ma-117	146	17	(	(	PUNCT
ma-117	146	18	1−	1−	NUM
ma-117	146	19	θ2∆t)2	θ2∆t)2	X
ma-117	147	1	+	+	CCONJ
ma-117	147	2	1	1	NUM
ma-117	147	3	]	]	PUNCT
ma-117	147	4	=	=	SYM
ma-117	147	5	(	(	PUNCT
ma-117	147	6	1−	1−	NUM
ma-117	147	7	θ2∆t)2k+1e	θ2∆t)2k+1e	PROPN
ma-117	147	8	(	(	PUNCT
ma-117	147	9	|x0|2	|x0|2	PROPN
ma-117	147	10	)	)	PUNCT
ma-117	147	11	+	+	CCONJ
ma-117	147	12	(	(	PUNCT
ma-117	147	13	θ2	θ2	ADV
ma-117	147	14	3	3	NUM
ma-117	147	15	+	+	CCONJ
ma-117	147	16	θ2	θ2	PROPN
ma-117	147	17	1∆t	1∆t	NUM
ma-117	147	18	)	)	PUNCT
ma-117	147	19	∆t	∆t	PROPN
ma-117	147	20	[	[	PUNCT
ma-117	147	21	(	(	PUNCT
ma-117	147	22	1−	1−	NUM
ma-117	147	23	θ2∆t)2k	θ2∆t)2k	PROPN
ma-117	147	24	+	+	NUM
ma-117	147	25	...	...	PUNCT
ma-117	148	1	+	+	CCONJ
ma-117	148	2	(	(	PUNCT
ma-117	148	3	1−	1−	NUM
ma-117	148	4	θ2∆t)4	θ2∆t)4	X
ma-117	148	5	+	+	PROPN
ma-117	148	6	(	(	PUNCT
ma-117	148	7	1−	1−	NUM
ma-117	148	8	θ2∆t)2	θ2∆t)2	X
ma-117	148	9	+	+	CCONJ
ma-117	148	10	(	(	PUNCT
ma-117	148	11	1−	1−	NUM
ma-117	148	12	θ2∆t)0	θ2∆t)0	NOUN
ma-117	148	13	]	]	PUNCT
ma-117	149	1	=	=	PUNCT
ma-117	149	2	(	(	PUNCT
ma-117	149	3	1−	1−	NUM
ma-117	149	4	θ2∆t)2(k+1	θ2∆t)2(k+1	NOUN
ma-117	149	5	)	)	PUNCT
ma-117	149	6	e	e	NOUN
ma-117	149	7	(	(	PUNCT
ma-117	149	8	|x0|2	|x0|2	PROPN
ma-117	149	9	)	)	PUNCT
ma-117	150	1	+	+	CCONJ
ma-117	150	2	(	(	PUNCT
ma-117	150	3	θ2	θ2	ADV
ma-117	150	4	3	3	NUM
ma-117	150	5	+	+	CCONJ
ma-117	150	6	θ2	θ2	PROPN
ma-117	150	7	1∆t	1∆t	NUM
ma-117	150	8	)	)	PUNCT
ma-117	150	9	∆t	∆t	PROPN
ma-117	150	10	k+1∑	k+1∑	PROPN
ma-117	151	1	i=0	i=0	PROPN
ma-117	151	2	(	(	PUNCT
ma-117	151	3	1−	1−	NUM
ma-117	151	4	θ2∆t)2i	θ2∆t)2i	NOUN
ma-117	151	5	=	=	SYM
ma-117	151	6	(	(	PUNCT
ma-117	151	7	1−	1−	NUM
ma-117	151	8	θ2∆t)2k+2	θ2∆t)2k+2	PROPN
ma-117	151	9	e	e	NOUN
ma-117	151	10	(	(	PUNCT
ma-117	151	11	|x0|2	|x0|2	PROPN
ma-117	151	12	)	)	PUNCT
ma-117	152	1	+	+	CCONJ
ma-117	152	2	(	(	PUNCT
ma-117	152	3	θ2	θ2	ADV
ma-117	152	4	3	3	NUM
ma-117	152	5	+	+	CCONJ
ma-117	152	6	θ2	θ2	PROPN
ma-117	152	7	1∆t	1∆t	NUM
ma-117	152	8	)	)	PUNCT
ma-117	152	9	∆t	∆t	PROPN
ma-117	153	1	[	[	PUNCT
ma-117	153	2	1−	1−	NUM
ma-117	153	3	|1−	|1−	PROPN
ma-117	153	4	θ2∆t|2k+2	θ2∆t|2k+2	NUM
ma-117	153	5	1−	1−	NUM
ma-117	153	6	|1−	|1−	NOUN
ma-117	153	7	θ2∆t|2	θ2∆t|2	VERB
ma-117	153	8	]	]	X
ma-117	153	9	we	we	PRON
ma-117	153	10	get	get	VERB
ma-117	153	11	:	:	PUNCT
ma-117	153	12	e	e	X
ma-117	154	1	[	[	X
ma-117	154	2	∣∣xemk+1	∣∣xemk+1	VERB
ma-117	154	3	∣∣2	∣∣2	NOUN
ma-117	154	4	]	]	X
ma-117	154	5	=	=	SYM
ma-117	154	6	(	(	PUNCT
ma-117	154	7	1−	1−	NUM
ma-117	154	8	θ2∆t)2k+2	θ2∆t)2k+2	PROPN
ma-117	154	9	e	e	NOUN
ma-117	154	10	(	(	PUNCT
ma-117	154	11	|x0|2	|x0|2	PROPN
ma-117	154	12	)	)	PUNCT
ma-117	155	1	+	+	CCONJ
ma-117	155	2	(	(	PUNCT
ma-117	155	3	θ2	θ2	ADV
ma-117	155	4	3	3	NUM
ma-117	155	5	+	+	CCONJ
ma-117	155	6	θ2	θ2	PROPN
ma-117	155	7	1∆t	1∆t	NUM
ma-117	155	8	)	)	PUNCT
ma-117	155	9	∆t	∆t	PROPN
ma-117	156	1	[	[	PUNCT
ma-117	156	2	1	1	NUM
ma-117	156	3	1−	1−	NUM
ma-117	156	4	|1−	|1−	NOUN
ma-117	156	5	θ2∆t|2	θ2∆t|2	VERB
ma-117	156	6	]	]	PUNCT
ma-117	156	7	(	(	PUNCT
ma-117	156	8	3.9	3.9	NUM
ma-117	156	9	)	)	PUNCT
ma-117	156	10	the	the	DET
ma-117	156	11	expression	expression	NOUN
ma-117	156	12	(	(	PUNCT
ma-117	156	13	3.9	3.9	NUM
ma-117	156	14	)	)	PUNCT
ma-117	156	15	as	as	ADP
ma-117	156	16	the	the	DET
ma-117	156	17	geometric	geometric	ADJ
ma-117	156	18	sequence	sequence	NOUN
ma-117	156	19	,	,	PUNCT
ma-117	156	20	we	we	PRON
ma-117	156	21	have	have	VERB
ma-117	156	22	that	that	SCONJ
ma-117	156	23	it	it	PRON
ma-117	156	24	converges	converge	VERB
ma-117	156	25	when	when	SCONJ
ma-117	156	26	|1−	|1−	VERB
ma-117	156	27	θ2∆t|	θ2∆t|	PROPN
ma-117	156	28	<	<	X
ma-117	156	29	1	1	NUM
ma-117	156	30	passing	pass	VERB
ma-117	156	31	to	to	ADP
ma-117	156	32	the	the	DET
ma-117	156	33	limit	limit	NOUN
ma-117	156	34	of	of	ADP
ma-117	156	35	the	the	DET
ma-117	156	36	equation	equation	NOUN
ma-117	156	37	(	(	PUNCT
ma-117	156	38	3.9	3.9	NUM
ma-117	156	39	)	)	PUNCT
ma-117	156	40	,	,	PUNCT
ma-117	156	41	for	for	ADP
ma-117	156	42	∆t	∆t	PROPN
ma-117	156	43	→	→	SYM
ma-117	156	44	0	0	NUM
ma-117	156	45	and	and	CCONJ
ma-117	156	46	k	k	PROPN
ma-117	156	47	→	→	SYM
ma-117	156	48	+	+	NOUN
ma-117	156	49	∞	∞	PROPN
ma-117	156	50	,	,	PUNCT
ma-117	156	51	we	we	PRON
ma-117	156	52	find	find	VERB
ma-117	156	53	the	the	DET
ma-117	156	54	desired	desire	VERB
ma-117	156	55	result	result	NOUN
ma-117	156	56	i.e	i.e	PROPN
ma-117	156	57	:	:	PUNCT
ma-117	156	58	lim	lim	PROPN
ma-117	156	59	∆t→0	∆t→0	PROPN
ma-117	156	60	(	(	PUNCT
ma-117	156	61	lim	lim	PROPN
ma-117	156	62	k→+∞	k→+∞	PROPN
ma-117	156	63	e	e	PROPN
ma-117	156	64	[	[	PUNCT
ma-117	156	65	xemk+1	xemk+1	X
ma-117	156	66	]	]	X
ma-117	156	67	)	)	PUNCT
ma-117	157	1	=	=	SYM
ma-117	157	2	θ2	θ2	ADP
ma-117	157	3	3	3	NUM
ma-117	157	4	2θ2	2θ2	NUM
ma-117	157	5	�	�	PROPN
ma-117	157	6	https://doi.org/10.28924/ada/ma.3.8	https://doi.org/10.28924/ada/ma.3.8	PROPN
ma-117	157	7	eur	eur	PROPN
ma-117	157	8	.	.	PUNCT
ma-117	158	1	j.	j.	PROPN
ma-117	158	2	math	math	PROPN
ma-117	158	3	.	.	PUNCT
ma-117	159	1	anal	anal	PROPN
ma-117	159	2	.	.	PUNCT
ma-117	160	1	10.28924	10.28924	NUM
ma-117	160	2	/	/	SYM
ma-117	160	3	ada	ada	PROPN
ma-117	160	4	/	/	SYM
ma-117	160	5	ma.3.8	ma.3.8	PROPN
ma-117	160	6	83.3	83.3	NUM
ma-117	160	7	.	.	PUNCT
ma-117	161	1	milshtein	milshtein	PROPN
ma-117	161	2	’s	’s	PART
ma-117	161	3	scheme	scheme	NOUN
ma-117	161	4	stabilities	stability	NOUN
ma-117	161	5	.	.	PUNCT
ma-117	162	1	the	the	DET
ma-117	162	2	milshtein	milshtein	PROPN
ma-117	162	3	scheme	scheme	NOUN
ma-117	162	4	associated	associate	VERB
ma-117	162	5	to	to	ADP
ma-117	162	6	the	the	DET
ma-117	162	7	system	system	NOUN
ma-117	162	8	(	(	PUNCT
ma-117	162	9	3.1	3.1	NUM
ma-117	162	10	)	)	PUNCT
ma-117	162	11	is	be	AUX
ma-117	162	12	:	:	PUNCT
ma-117	162	13	xmk+1	xmk+1	NOUN
ma-117	162	14	=	=	PUNCT
ma-117	162	15	θ1∆t	θ1∆t	NOUN
ma-117	163	1	+	+	CCONJ
ma-117	163	2	(	(	PUNCT
ma-117	163	3	1−	1−	NUM
ma-117	163	4	θ2∆t)xk	θ2∆t)xk	NOUN
ma-117	163	5	+	+	CCONJ
ma-117	163	6	θ3	θ3	NOUN
ma-117	163	7	√	√	PROPN
ma-117	163	8	∆tzk	∆tzk	NOUN
ma-117	163	9	(	(	PUNCT
ma-117	163	10	3.10	3.10	NUM
ma-117	163	11	)	)	PUNCT
ma-117	163	12	with	with	ADP
ma-117	163	13	σ	σ	PROPN
ma-117	163	14	=	=	SYM
ma-117	163	15	θ3	θ3	PROPN
ma-117	163	16	σ′	σ′	PROPN
ma-117	163	17	=	=	SYM
ma-117	163	18	0	0	PUNCT
ma-117	163	19	then	then	ADV
ma-117	163	20	mean	mean	VERB
ma-117	163	21	and	and	CCONJ
ma-117	163	22	mean	mean	ADJ
ma-117	163	23	-	-	PUNCT
ma-117	163	24	square	square	ADJ
ma-117	163	25	stabilities	stability	NOUN
ma-117	163	26	gives	give	VERB
ma-117	163	27	the	the	DET
ma-117	163	28	same	same	ADJ
ma-117	163	29	results	result	NOUN
ma-117	163	30	as	as	ADP
ma-117	163	31	in	in	ADP
ma-117	163	32	the	the	DET
ma-117	163	33	euler	euler	NOUN
ma-117	163	34	-	-	PUNCT
ma-117	163	35	maruyama	maruyama	NOUN
ma-117	163	36	schemei.e	schemei.e	ADV
ma-117	163	37	theorem	theorem	VERB
ma-117	163	38	3.3	3.3	NUM
ma-117	163	39	.	.	PUNCT
ma-117	164	1	(	(	PUNCT
ma-117	164	2	mean	mean	VERB
ma-117	164	3	stability	stability	NOUN
ma-117	164	4	of	of	ADP
ma-117	164	5	milshtein	milshtein	PROPN
ma-117	164	6	scheme	scheme	NOUN
ma-117	164	7	)	)	PUNCT
ma-117	164	8	the	the	DET
ma-117	164	9	milshtein	milshtein	PROPN
ma-117	164	10	scheme	scheme	NOUN
ma-117	164	11	(	(	PUNCT
ma-117	164	12	3.10	3.10	NUM
ma-117	164	13	)	)	PUNCT
ma-117	164	14	of	of	ADP
ma-117	164	15	vasicek	vasicek	PROPN
ma-117	164	16	model(3.1	model(3.1	PROPN
ma-117	164	17	)	)	PUNCT
ma-117	164	18	is	be	AUX
ma-117	164	19	mean	mean	VERB
ma-117	164	20	asymptotically	asymptotically	ADV
ma-117	164	21	stable	stable	ADJ
ma-117	164	22	if	if	SCONJ
ma-117	164	23	:	:	PUNCT
ma-117	164	24	e	e	X
ma-117	164	25	[	[	PUNCT
ma-117	164	26	xmk+1	xmk+1	X
ma-117	164	27	]	]	PUNCT
ma-117	165	1	=	=	PUNCT
ma-117	165	2	θ1∆t	θ1∆t	NOUN
ma-117	165	3	[	[	PUNCT
ma-117	165	4	k+1∑	k+1∑	PROPN
ma-117	165	5	i=0	i=0	PROPN
ma-117	165	6	(	(	PUNCT
ma-117	165	7	1−	1−	NUM
ma-117	165	8	θ2∆t)i	θ2∆t)i	X
ma-117	165	9	]	]	PUNCT
ma-117	166	1	+	+	CCONJ
ma-117	166	2	(	(	PUNCT
ma-117	166	3	1−	1−	NUM
ma-117	166	4	θ2∆t)k+1e[x0	θ2∆t)k+1e[x0	NOUN
ma-117	166	5	]	]	PUNCT
ma-117	166	6	and	and	CCONJ
ma-117	166	7	that	that	SCONJ
ma-117	166	8	the	the	DET
ma-117	166	9	following	follow	VERB
ma-117	166	10	two	two	NUM
ma-117	166	11	conditions	condition	NOUN
ma-117	166	12	are	be	AUX
ma-117	166	13	satisfied	satisfied	ADJ
ma-117	166	14	simultaneously	simultaneously	ADV
ma-117	166	15	:	:	PUNCT
ma-117	166	16	(	(	PUNCT
ma-117	166	17	1	1	X
ma-117	166	18	)	)	PUNCT
ma-117	166	19	|1−	|1−	VERB
ma-117	167	1	θ2∆t|	θ2∆t|	ADV
ma-117	167	2	<	<	X
ma-117	167	3	1	1	NUM
ma-117	167	4	(	(	PUNCT
ma-117	167	5	2	2	NUM
ma-117	167	6	)	)	PUNCT
ma-117	167	7	lim	lim	NOUN
ma-117	167	8	∆t→0	∆t→0	PROPN
ma-117	167	9	(	(	PUNCT
ma-117	167	10	lim	lim	PROPN
ma-117	167	11	k→+∞	k→+∞	PROPN
ma-117	167	12	e	e	PROPN
ma-117	167	13	[	[	PUNCT
ma-117	167	14	xmk+1	xmk+1	X
ma-117	167	15	]	]	PUNCT
ma-117	167	16	)	)	PUNCT
ma-117	167	17	=	=	SYM
ma-117	167	18	θ1	θ1	NOUN
ma-117	167	19	θ2	θ2	NOUN
ma-117	167	20	theorem	theorem	VERB
ma-117	167	21	3.4	3.4	NUM
ma-117	167	22	.	.	PUNCT
ma-117	168	1	(	(	PUNCT
ma-117	168	2	mean	mean	ADJ
ma-117	168	3	-	-	PUNCT
ma-117	168	4	square	square	ADJ
ma-117	168	5	stability	stability	NOUN
ma-117	168	6	of	of	ADP
ma-117	168	7	milshtein	milshtein	PROPN
ma-117	168	8	scheme	scheme	NOUN
ma-117	168	9	)	)	PUNCT
ma-117	168	10	the	the	DET
ma-117	168	11	milshtein	milshtein	PROPN
ma-117	168	12	scheme	scheme	NOUN
ma-117	168	13	(	(	PUNCT
ma-117	168	14	3.10	3.10	NUM
ma-117	168	15	)	)	PUNCT
ma-117	168	16	of	of	ADP
ma-117	168	17	vasicek	vasicek	PROPN
ma-117	168	18	model	model	NOUN
ma-117	168	19	(	(	PUNCT
ma-117	168	20	3.1	3.1	NUM
ma-117	168	21	)	)	PUNCT
ma-117	168	22	is	be	AUX
ma-117	168	23	mean	mean	ADJ
ma-117	168	24	-	-	PUNCT
ma-117	168	25	square	square	ADJ
ma-117	168	26	asymptotically	asymptotically	ADV
ma-117	168	27	stable	stable	ADJ
ma-117	168	28	if	if	SCONJ
ma-117	168	29	:	:	PUNCT
ma-117	168	30	e	e	X
ma-117	168	31	[	[	X
ma-117	168	32	∣∣xmk+1	∣∣xmk+1	X
ma-117	168	33	∣∣2	∣∣2	NUM
ma-117	168	34	]	]	X
ma-117	169	1	=	=	SYM
ma-117	169	2	(	(	PUNCT
ma-117	169	3	θ2	θ2	ADV
ma-117	169	4	3	3	NUM
ma-117	169	5	+	+	CCONJ
ma-117	169	6	θ2	θ2	PROPN
ma-117	169	7	1∆t	1∆t	NUM
ma-117	169	8	)	)	PUNCT
ma-117	169	9	∆t	∆t	PROPN
ma-117	170	1	k+1∑	k+1∑	PROPN
ma-117	170	2	i=0	i=0	PROPN
ma-117	170	3	(	(	PUNCT
ma-117	170	4	1−	1−	NUM
ma-117	170	5	θ2∆t)2i	θ2∆t)2i	X
ma-117	170	6	+	+	CCONJ
ma-117	170	7	(	(	PUNCT
ma-117	170	8	1−	1−	NUM
ma-117	170	9	θ2∆t)2(k+1	θ2∆t)2(k+1	NOUN
ma-117	170	10	)	)	PUNCT
ma-117	170	11	e	e	NOUN
ma-117	170	12	(	(	PUNCT
ma-117	170	13	|x0|2	|x0|2	PROPN
ma-117	170	14	)	)	PUNCT
ma-117	170	15	and	and	CCONJ
ma-117	170	16	that	that	SCONJ
ma-117	170	17	the	the	DET
ma-117	170	18	following	follow	VERB
ma-117	170	19	two	two	NUM
ma-117	170	20	conditions	condition	NOUN
ma-117	170	21	are	be	AUX
ma-117	170	22	satisfied	satisfied	ADJ
ma-117	170	23	simultaneously	simultaneously	ADV
ma-117	170	24	:	:	PUNCT
ma-117	170	25	(	(	PUNCT
ma-117	170	26	1	1	X
ma-117	170	27	)	)	PUNCT
ma-117	170	28	|1−	|1−	VERB
ma-117	171	1	θ2∆t|	θ2∆t|	ADV
ma-117	171	2	<	<	X
ma-117	171	3	1	1	NUM
ma-117	171	4	(	(	PUNCT
ma-117	171	5	2	2	NUM
ma-117	171	6	)	)	PUNCT
ma-117	171	7	lim	lim	NOUN
ma-117	171	8	∆t→0	∆t→0	PROPN
ma-117	171	9	(	(	PUNCT
ma-117	171	10	lim	lim	PROPN
ma-117	171	11	k→+∞	k→+∞	PROPN
ma-117	171	12	e	e	PROPN
ma-117	171	13	[	[	PUNCT
ma-117	171	14	xmk+1	xmk+1	X
ma-117	171	15	]	]	X
ma-117	171	16	2	2	X
ma-117	171	17	)	)	PUNCT
ma-117	171	18	=	=	SYM
ma-117	171	19	θ2	θ2	ADP
ma-117	171	20	3	3	NUM
ma-117	171	21	2θ2	2θ2	NUM
ma-117	171	22	proof	proof	NOUN
ma-117	171	23	.	.	PUNCT
ma-117	172	1	the	the	DET
ma-117	172	2	proofs	proof	NOUN
ma-117	172	3	of	of	ADP
ma-117	172	4	these	these	DET
ma-117	172	5	theorems	theorem	NOUN
ma-117	172	6	above	above	ADV
ma-117	172	7	is	be	AUX
ma-117	172	8	done	do	VERB
ma-117	172	9	in	in	ADP
ma-117	172	10	the	the	DET
ma-117	172	11	same	same	ADJ
ma-117	172	12	way	way	NOUN
ma-117	172	13	as	as	ADP
ma-117	172	14	the	the	DET
ma-117	172	15	result	result	NOUN
ma-117	172	16	theorems	theorem	NOUN
ma-117	172	17	of	of	ADP
ma-117	172	18	theeuler	theeuler	NOUN
ma-117	172	19	-	-	PUNCT
ma-117	172	20	maruyama	maruyama	NOUN
ma-117	172	21	scheme	scheme	NOUN
ma-117	172	22	for	for	ADP
ma-117	172	23	vasicek	vasicek	PROPN
ma-117	172	24	model	model	NOUN
ma-117	172	25	.	.	PUNCT
ma-117	173	1	�	�	PROPN
ma-117	173	2	3.4	3.4	NUM
ma-117	173	3	.	.	PUNCT
ma-117	174	1	implicit	implicit	ADJ
ma-117	174	2	euler	euler	VERB
ma-117	174	3	-	-	PUNCT
ma-117	174	4	maruyama	maruyama	NOUN
ma-117	174	5	scheme	scheme	NOUN
ma-117	174	6	stabilities	stability	NOUN
ma-117	174	7	.	.	PUNCT
ma-117	175	1	the	the	DET
ma-117	175	2	implicit	implicit	ADJ
ma-117	175	3	euler	euler	VERB
ma-117	175	4	-	-	PUNCT
ma-117	175	5	maruyama	maruyama	NOUN
ma-117	175	6	scheme	scheme	NOUN
ma-117	175	7	associ	associ	PROPN
ma-117	175	8	-	-	PUNCT
ma-117	175	9	ated	ate	VERB
ma-117	175	10	to	to	ADP
ma-117	175	11	the	the	DET
ma-117	175	12	system	system	NOUN
ma-117	175	13	(	(	PUNCT
ma-117	175	14	3.1	3.1	NUM
ma-117	175	15	)	)	PUNCT
ma-117	175	16	is	be	AUX
ma-117	175	17	:	:	PUNCT
ma-117	175	18	xiemk+1	xiemk+1	PROPN
ma-117	176	1	=	=	PUNCT
ma-117	176	2	xk	xk	PROPN
ma-117	177	1	+	+	CCONJ
ma-117	177	2	(	(	PUNCT
ma-117	177	3	θ1	θ1	NOUN
ma-117	177	4	−	−	NOUN
ma-117	177	5	θ2xk+1)∆t	θ2xk+1)∆t	NOUN
ma-117	177	6	+	+	CCONJ
ma-117	177	7	θ3	θ3	NOUN
ma-117	177	8	√	√	PROPN
ma-117	177	9	∆tzk	∆tzk	NOUN
ma-117	177	10	xk+1	xk+1	NUM
ma-117	178	1	+	+	CCONJ
ma-117	178	2	θ2∆txk+1	θ2∆txk+1	NOUN
ma-117	178	3	=	=	SYM
ma-117	178	4	xk	xk	X
ma-117	178	5	+	+	PUNCT
ma-117	178	6	θ1∆t	θ1∆t	NOUN
ma-117	178	7	+	+	SYM
ma-117	178	8	θ3	θ3	NOUN
ma-117	178	9	√	√	PROPN
ma-117	178	10	∆tzk	∆tzk	NOUN
ma-117	178	11	xk+1	xk+1	NUM
ma-117	178	12	(	(	PUNCT
ma-117	178	13	1	1	NUM
ma-117	178	14	+	+	NUM
ma-117	178	15	θ2∆t	θ2∆t	NOUN
ma-117	178	16	)	)	PUNCT
ma-117	178	17	=	=	PUNCT
ma-117	178	18	θ1∆t	θ1∆t	NOUN
ma-117	178	19	+	+	ADJ
ma-117	178	20	xk	xk	NOUN
ma-117	178	21	+	+	CCONJ
ma-117	178	22	θ3	θ3	PROPN
ma-117	178	23	√	√	PROPN
ma-117	178	24	∆tzk	∆tzk	NOUN
ma-117	178	25	xk+1	xk+1	NUM
ma-117	178	26	=	=	PUNCT
ma-117	178	27	θ1∆t	θ1∆t	NOUN
ma-117	178	28	1	1	NUM
ma-117	178	29	+	+	NUM
ma-117	178	30	θ2∆t	θ2∆t	NOUN
ma-117	178	31	+	+	CCONJ
ma-117	178	32	1	1	NUM
ma-117	178	33	1	1	NUM
ma-117	178	34	+	+	NUM
ma-117	178	35	θ2∆t	θ2∆t	NOUN
ma-117	178	36	xk	xk	NOUN
ma-117	179	1	+	+	CCONJ
ma-117	179	2	θ3	θ3	PROPN
ma-117	179	3	√	√	ADV
ma-117	179	4	∆t	∆t	PROPN
ma-117	179	5	1	1	NUM
ma-117	179	6	+	+	NUM
ma-117	179	7	θ2∆t	θ2∆t	NOUN
ma-117	179	8	zk	zk	PROPN
ma-117	179	9	https://doi.org/10.28924/ada/ma.3.8	https://doi.org/10.28924/ada/ma.3.8	PROPN
ma-117	179	10	eur	eur	PROPN
ma-117	179	11	.	.	PUNCT
ma-117	180	1	j.	j.	PROPN
ma-117	180	2	math	math	PROPN
ma-117	180	3	.	.	PUNCT
ma-117	181	1	anal	anal	PROPN
ma-117	181	2	.	.	PUNCT
ma-117	182	1	10.28924	10.28924	NUM
ma-117	182	2	/	/	SYM
ma-117	182	3	ada	ada	PROPN
ma-117	182	4	/	/	SYM
ma-117	182	5	ma.3.8	ma.3.8	PROPN
ma-117	182	6	9we	9we	NOUN
ma-117	182	7	get	get	VERB
ma-117	182	8	:	:	PUNCT
ma-117	182	9	xiemk+1	xiemk+1	PUNCT
ma-117	182	10	=	=	PRON
ma-117	182	11	θ1∆t	θ1∆t	NOUN
ma-117	182	12	1	1	NUM
ma-117	183	1	+	+	NUM
ma-117	183	2	θ2∆t	θ2∆t	NOUN
ma-117	183	3	+	+	CCONJ
ma-117	183	4	1	1	NUM
ma-117	183	5	1	1	NUM
ma-117	183	6	+	+	NUM
ma-117	183	7	θ2∆t	θ2∆t	NOUN
ma-117	183	8	xk	xk	NOUN
ma-117	184	1	+	+	CCONJ
ma-117	184	2	θ3	θ3	PROPN
ma-117	184	3	√	√	ADV
ma-117	184	4	∆t	∆t	PROPN
ma-117	184	5	1	1	NUM
ma-117	184	6	+	+	NUM
ma-117	184	7	θ2∆t	θ2∆t	NOUN
ma-117	184	8	zk	zk	PROPN
ma-117	184	9	(	(	PUNCT
ma-117	184	10	3.11	3.11	NUM
ma-117	184	11	)	)	PUNCT
ma-117	184	12	3.4.1	3.4.1	NUM
ma-117	184	13	.	.	PUNCT
ma-117	185	1	mean	mean	VERB
ma-117	185	2	stability	stability	NOUN
ma-117	185	3	of	of	ADP
ma-117	185	4	implicit	implicit	ADJ
ma-117	185	5	euler	euler	NOUN
ma-117	185	6	-	-	PUNCT
ma-117	185	7	maruyama	maruyama	NOUN
ma-117	185	8	scheme	scheme	NOUN
ma-117	185	9	.	.	PUNCT
ma-117	186	1	theorem	theorem	VERB
ma-117	186	2	3.5	3.5	NUM
ma-117	186	3	.	.	PUNCT
ma-117	187	1	(	(	PUNCT
ma-117	187	2	mean	mean	VERB
ma-117	187	3	stability	stability	NOUN
ma-117	187	4	of	of	ADP
ma-117	187	5	implicit	implicit	ADJ
ma-117	187	6	euler	euler	NOUN
ma-117	187	7	-	-	PUNCT
ma-117	187	8	maruyama	maruyama	NOUN
ma-117	187	9	scheme	scheme	NOUN
ma-117	187	10	)	)	PUNCT
ma-117	187	11	the	the	DET
ma-117	187	12	implicit	implicit	ADJ
ma-117	187	13	euler	euler	VERB
ma-117	187	14	-	-	PUNCT
ma-117	187	15	maruyama	maruyama	NOUN
ma-117	187	16	scheme	scheme	NOUN
ma-117	187	17	(	(	PUNCT
ma-117	187	18	3.11	3.11	NUM
ma-117	187	19	)	)	PUNCT
ma-117	187	20	of	of	ADP
ma-117	187	21	vasicek	vasicek	PROPN
ma-117	187	22	model	model	NOUN
ma-117	187	23	(	(	PUNCT
ma-117	187	24	3.1	3.1	NUM
ma-117	187	25	)	)	PUNCT
ma-117	187	26	is	be	AUX
ma-117	187	27	mean	mean	VERB
ma-117	187	28	asymptotically	asymptotically	ADV
ma-117	187	29	stable	stable	ADJ
ma-117	187	30	if	if	SCONJ
ma-117	187	31	:	:	PUNCT
ma-117	188	1	e	e	X
ma-117	188	2	(	(	PUNCT
ma-117	188	3	xiemk+1	xiemk+1	X
ma-117	188	4	)	)	PUNCT
ma-117	189	1	=	=	PUNCT
ma-117	189	2	(	(	PUNCT
ma-117	189	3	1	1	NUM
ma-117	189	4	1	1	NUM
ma-117	189	5	+	+	NUM
ma-117	189	6	θ2∆t	θ2∆t	NOUN
ma-117	189	7	)	)	PUNCT
ma-117	189	8	k+1	k+1	X
ma-117	189	9	e	e	X
ma-117	189	10	(	(	PUNCT
ma-117	189	11	x0	x0	PROPN
ma-117	189	12	)	)	PUNCT
ma-117	190	1	+	+	CCONJ
ma-117	190	2	θ1∆t	θ1∆t	NOUN
ma-117	190	3	k+1∑	k+1∑	PROPN
ma-117	190	4	i=0	i=0	PROPN
ma-117	190	5	(	(	PUNCT
ma-117	190	6	1	1	NUM
ma-117	190	7	1	1	NUM
ma-117	190	8	+	+	NUM
ma-117	190	9	θ2∆t	θ2∆t	NOUN
ma-117	190	10	)	)	PUNCT
ma-117	191	1	i	i	PRON
ma-117	191	2	and	and	CCONJ
ma-117	191	3	that	that	SCONJ
ma-117	191	4	the	the	DET
ma-117	191	5	following	follow	VERB
ma-117	191	6	two	two	NUM
ma-117	191	7	conditions	condition	NOUN
ma-117	191	8	are	be	AUX
ma-117	191	9	satisfied	satisfied	ADJ
ma-117	191	10	simultaneously	simultaneously	ADV
ma-117	191	11	:	:	PUNCT
ma-117	191	12	(	(	PUNCT
ma-117	191	13	1	1	X
ma-117	191	14	)	)	PUNCT
ma-117	191	15	|1	|1	PUNCT
ma-117	192	1	+	+	CCONJ
ma-117	193	1	θ2∆t|	θ2∆t|	ADP
ma-117	193	2	>	>	SYM
ma-117	193	3	1	1	NUM
ma-117	193	4	(	(	PUNCT
ma-117	193	5	2	2	NUM
ma-117	193	6	)	)	PUNCT
ma-117	193	7	lim	lim	PROPN
ma-117	193	8	∆t→0	∆t→0	PROPN
ma-117	193	9	(	(	PUNCT
ma-117	193	10	lim	lim	PROPN
ma-117	193	11	k→∞	k→∞	NOUN
ma-117	193	12	e	e	PROPN
ma-117	193	13	[	[	PUNCT
ma-117	193	14	xiemk+1	xiemk+1	X
ma-117	193	15	]	]	X
ma-117	193	16	)	)	PUNCT
ma-117	193	17	=	=	SYM
ma-117	193	18	θ1	θ1	NOUN
ma-117	193	19	θ2	θ2	ADP
ma-117	193	20	proof	proof	NOUN
ma-117	193	21	.	.	PUNCT
ma-117	194	1	we	we	PRON
ma-117	194	2	start	start	VERB
ma-117	194	3	by	by	ADP
ma-117	194	4	evaluating	evaluate	VERB
ma-117	194	5	the	the	DET
ma-117	194	6	mean	mean	NOUN
ma-117	194	7	of	of	ADP
ma-117	194	8	the	the	DET
ma-117	194	9	implicit	implicit	ADJ
ma-117	194	10	euler	euler	NOUN
ma-117	194	11	-	-	PUNCT
ma-117	194	12	maruyama	maruyama	NOUN
ma-117	194	13	scheme	scheme	NOUN
ma-117	194	14	of	of	ADP
ma-117	194	15	the	the	DET
ma-117	194	16	expressiondefined	expressiondefine	VERB
ma-117	194	17	in	in	ADP
ma-117	194	18	(	(	PUNCT
ma-117	194	19	3.11	3.11	NUM
ma-117	194	20	)	)	PUNCT
ma-117	194	21	,	,	PUNCT
ma-117	194	22	in	in	ADP
ma-117	194	23	effect	effect	NOUN
ma-117	194	24	:	:	PUNCT
ma-117	194	25	e	e	X
ma-117	194	26	(	(	PUNCT
ma-117	194	27	xiemk+1	xiemk+1	X
ma-117	194	28	)	)	PUNCT
ma-117	195	1	=	=	SYM
ma-117	195	2	e	e	X
ma-117	195	3	(	(	PUNCT
ma-117	195	4	θ1∆t	θ1∆t	NOUN
ma-117	195	5	1	1	NUM
ma-117	195	6	+	+	NUM
ma-117	195	7	θ2∆t	θ2∆t	NOUN
ma-117	195	8	)	)	PUNCT
ma-117	196	1	+	+	CCONJ
ma-117	196	2	e	e	X
ma-117	196	3	(	(	PUNCT
ma-117	196	4	1	1	NUM
ma-117	196	5	1	1	NUM
ma-117	196	6	+	+	NUM
ma-117	196	7	θ2∆t	θ2∆t	NOUN
ma-117	196	8	xk	xk	NOUN
ma-117	196	9	)	)	PUNCT
ma-117	197	1	+	+	CCONJ
ma-117	197	2	e	e	X
ma-117	197	3	(	(	PUNCT
ma-117	197	4	θ3	θ3	ADV
ma-117	197	5	√	√	ADV
ma-117	197	6	∆t	∆t	PROPN
ma-117	197	7	1	1	NUM
ma-117	197	8	+	+	NUM
ma-117	197	9	θ2∆t	θ2∆t	NOUN
ma-117	197	10	zk	zk	NOUN
ma-117	197	11	)	)	PUNCT
ma-117	198	1	=	=	PRON
ma-117	198	2	θ1∆t	θ1∆t	NOUN
ma-117	198	3	1	1	NUM
ma-117	199	1	+	+	NUM
ma-117	199	2	θ2∆t	θ2∆t	NOUN
ma-117	199	3	+	+	CCONJ
ma-117	200	1	1	1	NUM
ma-117	200	2	1	1	NUM
ma-117	200	3	+	+	NUM
ma-117	200	4	θ2∆t	θ2∆t	NOUN
ma-117	200	5	e	e	NOUN
ma-117	200	6	(	(	PUNCT
ma-117	200	7	xk	xk	PROPN
ma-117	200	8	)	)	PUNCT
ma-117	200	9	=	=	PRON
ma-117	200	10	θ1∆t	θ1∆t	NOUN
ma-117	200	11	1	1	NUM
ma-117	201	1	+	+	NUM
ma-117	201	2	θ2∆t	θ2∆t	NOUN
ma-117	201	3	+	+	CCONJ
ma-117	201	4	1	1	NUM
ma-117	201	5	1	1	NUM
ma-117	201	6	+	+	NUM
ma-117	201	7	θ2∆t	θ2∆t	NOUN
ma-117	201	8	{	{	PUNCT
ma-117	201	9	(	(	PUNCT
ma-117	201	10	1	1	NUM
ma-117	201	11	1	1	NUM
ma-117	201	12	+	+	NUM
ma-117	201	13	θ2∆t	θ2∆t	NOUN
ma-117	201	14	)	)	PUNCT
ma-117	202	1	e	e	NOUN
ma-117	202	2	(	(	PUNCT
ma-117	202	3	xk−1	xk−1	PROPN
ma-117	202	4	)	)	PUNCT
ma-117	203	1	+	+	CCONJ
ma-117	203	2	θ1∆t	θ1∆t	NOUN
ma-117	203	3	1	1	NUM
ma-117	203	4	+	+	NUM
ma-117	203	5	θ2∆t	θ2∆t	NOUN
ma-117	203	6	}	}	PUNCT
ma-117	203	7	=	=	PUNCT
ma-117	203	8	θ1∆t	θ1∆t	NOUN
ma-117	203	9	1	1	NUM
ma-117	203	10	+	+	NUM
ma-117	203	11	θ2∆t	θ2∆t	NOUN
ma-117	203	12	+	+	NOUN
ma-117	203	13	θ1∆t	θ1∆t	NOUN
ma-117	203	14	(	(	PUNCT
ma-117	203	15	1	1	NUM
ma-117	203	16	+	+	NUM
ma-117	203	17	θ2∆t)2	θ2∆t)2	X
ma-117	204	1	+	+	CCONJ
ma-117	204	2	1	1	NUM
ma-117	204	3	(	(	PUNCT
ma-117	204	4	1	1	NUM
ma-117	204	5	+	+	NUM
ma-117	204	6	θ2∆t)2e	θ2∆t)2e	NOUN
ma-117	204	7	(	(	PUNCT
ma-117	204	8	xk−1	xk−1	PROPN
ma-117	204	9	)	)	PUNCT
ma-117	204	10	=	=	PUNCT
ma-117	205	1	(	(	PUNCT
ma-117	205	2	1	1	NUM
ma-117	205	3	1	1	NUM
ma-117	205	4	+	+	NUM
ma-117	205	5	θ2∆t	θ2∆t	NOUN
ma-117	205	6	)	)	PUNCT
ma-117	205	7	2	2	NUM
ma-117	205	8	e	e	X
ma-117	205	9	(	(	PUNCT
ma-117	205	10	xk−1	xk−1	PROPN
ma-117	205	11	)	)	PUNCT
ma-117	206	1	+	+	CCONJ
ma-117	206	2	θ1∆t	θ1∆t	NOUN
ma-117	206	3	(	(	PUNCT
ma-117	206	4	1	1	NUM
ma-117	206	5	(	(	PUNCT
ma-117	206	6	1	1	NUM
ma-117	206	7	+	+	NUM
ma-117	206	8	θ2∆t)2	θ2∆t)2	X
ma-117	207	1	+	+	CCONJ
ma-117	207	2	1	1	NUM
ma-117	207	3	(	(	PUNCT
ma-117	207	4	1	1	NUM
ma-117	207	5	+	+	NUM
ma-117	207	6	θ2∆t	θ2∆t	NOUN
ma-117	207	7	)	)	PUNCT
ma-117	207	8	)	)	PUNCT
ma-117	208	1	=	=	PUNCT
ma-117	208	2	(	(	PUNCT
ma-117	208	3	1	1	NUM
ma-117	208	4	1	1	NUM
ma-117	208	5	+	+	NUM
ma-117	208	6	θ2∆t	θ2∆t	NOUN
ma-117	208	7	)	)	PUNCT
ma-117	208	8	3	3	NUM
ma-117	208	9	e	e	X
ma-117	208	10	(	(	PUNCT
ma-117	208	11	xk−2	xk−2	PROPN
ma-117	208	12	)	)	PUNCT
ma-117	209	1	+	+	CCONJ
ma-117	209	2	θ1∆t	θ1∆t	NOUN
ma-117	209	3	(	(	PUNCT
ma-117	209	4	(	(	PUNCT
ma-117	209	5	1	1	NUM
ma-117	209	6	1	1	NUM
ma-117	209	7	+	+	NUM
ma-117	209	8	θ2∆t	θ2∆t	NOUN
ma-117	209	9	)	)	PUNCT
ma-117	209	10	3	3	NUM
ma-117	210	1	+	+	CCONJ
ma-117	210	2	(	(	PUNCT
ma-117	210	3	1	1	NUM
ma-117	210	4	1	1	NUM
ma-117	210	5	+	+	NUM
ma-117	210	6	θ2∆t	θ2∆t	NOUN
ma-117	210	7	)	)	PUNCT
ma-117	210	8	2	2	NUM
ma-117	211	1	+	+	CCONJ
ma-117	211	2	(	(	PUNCT
ma-117	211	3	1	1	NUM
ma-117	211	4	1	1	NUM
ma-117	211	5	+	+	NUM
ma-117	211	6	θ2∆t	θ2∆t	NOUN
ma-117	211	7	)	)	PUNCT
ma-117	211	8	)	)	PUNCT
ma-117	212	1	=	=	PUNCT
ma-117	212	2	(	(	PUNCT
ma-117	212	3	1	1	NUM
ma-117	212	4	1	1	NUM
ma-117	212	5	+	+	NUM
ma-117	212	6	θ2∆t	θ2∆t	NOUN
ma-117	212	7	)	)	PUNCT
ma-117	212	8	4	4	NUM
ma-117	212	9	e	e	NOUN
ma-117	212	10	(	(	PUNCT
ma-117	212	11	xk−3	xk−3	PROPN
ma-117	212	12	)	)	PUNCT
ma-117	213	1	+	+	CCONJ
ma-117	213	2	θ1∆t	θ1∆t	NOUN
ma-117	213	3	(	(	PUNCT
ma-117	213	4	(	(	PUNCT
ma-117	213	5	1	1	NUM
ma-117	213	6	1	1	NUM
ma-117	213	7	+	+	NUM
ma-117	213	8	θ2∆t	θ2∆t	NOUN
ma-117	213	9	)	)	PUNCT
ma-117	213	10	4	4	NUM
ma-117	214	1	+	+	CCONJ
ma-117	214	2	(	(	PUNCT
ma-117	214	3	1	1	NUM
ma-117	214	4	1	1	NUM
ma-117	214	5	+	+	NUM
ma-117	214	6	θ2∆t	θ2∆t	NOUN
ma-117	214	7	)	)	PUNCT
ma-117	214	8	3	3	NUM
ma-117	214	9	+	+	CCONJ
ma-117	214	10	·	·	PUNCT
ma-117	214	11	·	·	PUNCT
ma-117	214	12	·	·	PUNCT
ma-117	214	13	+	+	NUM
ma-117	214	14	1	1	NUM
ma-117	214	15	)	)	PUNCT
ma-117	214	16	by	by	ADP
ma-117	214	17	continuing	continue	VERB
ma-117	214	18	the	the	DET
ma-117	214	19	iterations	iteration	NOUN
ma-117	214	20	until	until	ADP
ma-117	214	21	k	k	PROPN
ma-117	214	22	+	+	PROPN
ma-117	214	23	1	1	NUM
ma-117	214	24	,	,	PUNCT
ma-117	214	25	we	we	PRON
ma-117	214	26	obtain	obtain	VERB
ma-117	214	27	:	:	PUNCT
ma-117	214	28	e	e	X
ma-117	214	29	(	(	PUNCT
ma-117	214	30	xiemk+1	xiemk+1	X
ma-117	214	31	)	)	PUNCT
ma-117	215	1	=	=	PUNCT
ma-117	215	2	(	(	PUNCT
ma-117	215	3	1	1	NUM
ma-117	215	4	1	1	NUM
ma-117	215	5	+	+	NUM
ma-117	215	6	θ2∆t	θ2∆t	NOUN
ma-117	215	7	)	)	PUNCT
ma-117	215	8	k+1	k+1	X
ma-117	215	9	e	e	X
ma-117	215	10	(	(	PUNCT
ma-117	215	11	x0	x0	PROPN
ma-117	215	12	)	)	PUNCT
ma-117	216	1	+	+	CCONJ
ma-117	216	2	θ1∆t	θ1∆t	NOUN
ma-117	216	3	k+1∑	k+1∑	PROPN
ma-117	216	4	i=0	i=0	PROPN
ma-117	216	5	(	(	PUNCT
ma-117	216	6	1	1	NUM
ma-117	216	7	1	1	NUM
ma-117	216	8	+	+	NUM
ma-117	216	9	θ2∆t	θ2∆t	NOUN
ma-117	216	10	)	)	PUNCT
ma-117	217	1	i	i	PRON
ma-117	217	2	(	(	PUNCT
ma-117	217	3	3.12	3.12	NUM
ma-117	217	4	)	)	PUNCT
ma-117	217	5	the	the	DET
ma-117	217	6	equation	equation	NOUN
ma-117	217	7	(	(	PUNCT
ma-117	217	8	3.12	3.12	NUM
ma-117	217	9	)	)	PUNCT
ma-117	217	10	is	be	AUX
ma-117	217	11	the	the	DET
ma-117	217	12	geometric	geometric	ADJ
ma-117	217	13	sum	sum	NOUN
ma-117	217	14	of	of	ADP
ma-117	217	15	geometric	geometric	ADJ
ma-117	217	16	sequence	sequence	NOUN
ma-117	217	17	and	and	CCONJ
ma-117	217	18	geometric	geometric	ADJ
ma-117	217	19	series	series	NOUN
ma-117	217	20	,	,	PUNCT
ma-117	217	21	the	the	DET
ma-117	217	22	expres	expres	PROPN
ma-117	217	23	-	-	PUNCT
ma-117	217	24	sion	sion	PROPN
ma-117	217	25	:	:	PUNCT
ma-117	217	26	(	(	PUNCT
ma-117	217	27	1	1	NUM
ma-117	217	28	1	1	NUM
ma-117	217	29	+	+	NUM
ma-117	217	30	θ2∆t	θ2∆t	NOUN
ma-117	217	31	)	)	PUNCT
ma-117	217	32	<	<	X
ma-117	218	1	1	1	NUM
ma-117	218	2	https://doi.org/10.28924/ada/ma.3.8	https://doi.org/10.28924/ada/ma.3.8	PROPN
ma-117	218	3	eur	eur	NOUN
ma-117	218	4	.	.	PUNCT
ma-117	219	1	j.	j.	PROPN
ma-117	219	2	math	math	PROPN
ma-117	219	3	.	.	PUNCT
ma-117	220	1	anal	anal	PROPN
ma-117	220	2	.	.	PUNCT
ma-117	221	1	10.28924	10.28924	NUM
ma-117	221	2	/	/	SYM
ma-117	221	3	ada	ada	PROPN
ma-117	221	4	/	/	SYM
ma-117	221	5	ma.3.8	ma.3.8	PROPN
ma-117	221	6	10or	10or	NOUN
ma-117	221	7	|1	|1	PRON
ma-117	222	1	+	+	CCONJ
ma-117	222	2	θ2∆t|	θ2∆t|	ADJ
ma-117	222	3	>	>	X
ma-117	222	4	1by	1by	ADJ
ma-117	222	5	using	use	VERB
ma-117	222	6	limit	limit	NOUN
ma-117	222	7	of	of	ADP
ma-117	222	8	(	(	PUNCT
ma-117	222	9	3.12	3.12	NUM
ma-117	222	10	)	)	PUNCT
ma-117	222	11	,	,	PUNCT
ma-117	222	12	for	for	ADP
ma-117	222	13	∆t	∆t	PROPN
ma-117	222	14	→	→	SYM
ma-117	222	15	0	0	NUM
ma-117	222	16	and	and	CCONJ
ma-117	222	17	k	k	PROPN
ma-117	222	18	→	→	SYM
ma-117	222	19	+	+	NOUN
ma-117	222	20	∞	∞	PROPN
ma-117	222	21	,	,	PUNCT
ma-117	222	22	we	we	PRON
ma-117	222	23	get	get	VERB
ma-117	222	24	:	:	PUNCT
ma-117	222	25	lim	lim	PROPN
ma-117	222	26	∆t→0	∆t→0	PROPN
ma-117	222	27	(	(	PUNCT
ma-117	222	28	lim	lim	PROPN
ma-117	222	29	k→+∞	k→+∞	PROPN
ma-117	222	30	e	e	PROPN
ma-117	222	31	[	[	PUNCT
ma-117	222	32	xiemk+1	xiemk+1	X
ma-117	222	33	]	]	X
ma-117	222	34	)	)	PUNCT
ma-117	223	1	=	=	SYM
ma-117	223	2	θ1	θ1	PROPN
ma-117	223	3	θ2	θ2	PROPN
ma-117	223	4	�	�	PROPN
ma-117	223	5	3.4.2	3.4.2	NUM
ma-117	223	6	.	.	PUNCT
ma-117	224	1	mean	mean	ADJ
ma-117	224	2	-	-	PUNCT
ma-117	224	3	square	square	ADJ
ma-117	224	4	stability	stability	NOUN
ma-117	224	5	of	of	ADP
ma-117	224	6	implicit	implicit	ADJ
ma-117	224	7	euler	euler	NOUN
ma-117	224	8	-	-	PUNCT
ma-117	224	9	maruyama	maruyama	NOUN
ma-117	224	10	scheme	scheme	NOUN
ma-117	224	11	.	.	PUNCT
ma-117	225	1	theorem	theorem	VERB
ma-117	225	2	3.6	3.6	NUM
ma-117	225	3	.	.	PUNCT
ma-117	226	1	(	(	PUNCT
ma-117	226	2	mean	mean	ADJ
ma-117	226	3	-	-	PUNCT
ma-117	226	4	square	square	ADJ
ma-117	226	5	stability	stability	NOUN
ma-117	226	6	of	of	ADP
ma-117	226	7	implicit	implicit	ADJ
ma-117	226	8	euler	euler	NOUN
ma-117	226	9	-	-	PUNCT
ma-117	226	10	maruyama	maruyama	NOUN
ma-117	226	11	scheme	scheme	NOUN
ma-117	226	12	)	)	PUNCT
ma-117	226	13	the	the	DET
ma-117	226	14	implicit	implicit	ADJ
ma-117	226	15	eulermaruyama	eulermaruyama	NOUN
ma-117	226	16	of	of	ADP
ma-117	226	17	vasicek	vasicek	PROPN
ma-117	226	18	model	model	NOUN
ma-117	226	19	(	(	PUNCT
ma-117	226	20	3.1	3.1	NUM
ma-117	226	21	)	)	PUNCT
ma-117	226	22	is	be	AUX
ma-117	226	23	mean	mean	ADJ
ma-117	226	24	-	-	PUNCT
ma-117	226	25	square	square	ADJ
ma-117	226	26	asymptotically	asymptotically	ADV
ma-117	226	27	stable	stable	ADJ
ma-117	226	28	if	if	SCONJ
ma-117	226	29	:	:	PUNCT
ma-117	226	30	e	e	X
ma-117	226	31	(	(	PUNCT
ma-117	226	32	∣∣xiemk+1	∣∣xiemk+1	VERB
ma-117	226	33	∣∣2	∣∣2	NUM
ma-117	226	34	)	)	PUNCT
ma-117	226	35	=	=	PUNCT
ma-117	227	1	(	(	PUNCT
ma-117	227	2	1	1	NUM
ma-117	227	3	1	1	NUM
ma-117	227	4	+	+	NUM
ma-117	227	5	θ2∆t	θ2∆t	NOUN
ma-117	227	6	)	)	PUNCT
ma-117	227	7	2(k+1	2(k+1	NUM
ma-117	227	8	)	)	PUNCT
ma-117	227	9	e	e	NOUN
ma-117	227	10	(	(	PUNCT
ma-117	227	11	|x0|)2	|x0|)2	NOUN
ma-117	227	12	+	+	CCONJ
ma-117	228	1	(	(	PUNCT
ma-117	228	2	θ2	θ2	ADV
ma-117	228	3	3	3	NUM
ma-117	228	4	+	+	CCONJ
ma-117	228	5	θ2	θ2	PROPN
ma-117	228	6	1∆t	1∆t	NUM
ma-117	228	7	)	)	PUNCT
ma-117	228	8	∆t	∆t	PROPN
ma-117	228	9	k+1∑	k+1∑	PROPN
ma-117	228	10	i=0	i=0	PROPN
ma-117	228	11	(	(	PUNCT
ma-117	228	12	1	1	NUM
ma-117	228	13	1	1	NUM
ma-117	228	14	+	+	NUM
ma-117	228	15	θ2∆t	θ2∆t	NOUN
ma-117	228	16	)	)	PUNCT
ma-117	228	17	2i	2i	NOUN
ma-117	228	18	,	,	PUNCT
ma-117	228	19	with	with	ADP
ma-117	228	20	|1	|1	PRON
ma-117	228	21	+	+	CCONJ
ma-117	228	22	θ2∆t|	θ2∆t|	PROPN
ma-117	228	23	>	>	SYM
ma-117	228	24	1	1	NUM
ma-117	228	25	and	and	CCONJ
ma-117	228	26	lim	lim	PROPN
ma-117	228	27	∆t→0	∆t→0	PROPN
ma-117	228	28	(	(	PUNCT
ma-117	228	29	lim	lim	PROPN
ma-117	228	30	k→∞	k→∞	PROPN
ma-117	228	31	e	e	PROPN
ma-117	228	32	(	(	PUNCT
ma-117	228	33	|xk+1|2	|xk+1|2	PROPN
ma-117	228	34	)	)	PUNCT
ma-117	228	35	)	)	PUNCT
ma-117	229	1	=	=	SYM
ma-117	229	2	θ2	θ2	ADP
ma-117	229	3	3	3	NUM
ma-117	229	4	2θ2	2θ2	NUM
ma-117	229	5	proof	proof	NOUN
ma-117	229	6	.	.	PUNCT
ma-117	230	1	let	let	VERB
ma-117	230	2	us	we	PRON
ma-117	230	3	evaluate	evaluate	VERB
ma-117	230	4	the	the	DET
ma-117	230	5	mean	mean	ADJ
ma-117	230	6	-	-	PUNCT
ma-117	230	7	square	square	NOUN
ma-117	230	8	of	of	ADP
ma-117	230	9	the	the	DET
ma-117	230	10	implicit	implicit	ADJ
ma-117	230	11	euler	euler	NOUN
ma-117	230	12	-	-	PUNCT
ma-117	230	13	maruyama	maruyama	NOUN
ma-117	230	14	scheme	scheme	NOUN
ma-117	230	15	(	(	PUNCT
ma-117	230	16	3.11	3.11	NUM
ma-117	230	17	)	)	PUNCT
ma-117	230	18	,	,	PUNCT
ma-117	230	19	in	in	ADP
ma-117	230	20	effect	effect	NOUN
ma-117	230	21	:	:	PUNCT
ma-117	230	22	e	e	X
ma-117	230	23	(	(	PUNCT
ma-117	230	24	∣∣xiemk+1	∣∣xiemk+1	X
ma-117	230	25	∣∣2	∣∣2	NUM
ma-117	230	26	)	)	PUNCT
ma-117	230	27	=	=	SYM
ma-117	230	28	e	e	X
ma-117	230	29	(	(	PUNCT
ma-117	230	30	∣∣∣∣	∣∣∣∣	NOUN
ma-117	230	31	θ1∆t	θ1∆t	NOUN
ma-117	230	32	1	1	NUM
ma-117	230	33	+	+	NUM
ma-117	230	34	θ2∆t	θ2∆t	NOUN
ma-117	230	35	∣∣∣∣2	∣∣∣∣2	NOUN
ma-117	230	36	)	)	PUNCT
ma-117	231	1	+	+	CCONJ
ma-117	231	2	e	e	X
ma-117	231	3	(	(	PUNCT
ma-117	231	4	∣∣∣∣	∣∣∣∣	NOUN
ma-117	231	5	1	1	NUM
ma-117	231	6	1	1	NUM
ma-117	231	7	+	+	NUM
ma-117	231	8	θ2∆t	θ2∆t	NOUN
ma-117	231	9	xk	xk	PROPN
ma-117	231	10	∣∣∣∣)+	∣∣∣∣)+	NUM
ma-117	231	11	∣∣∣∣	∣∣∣∣	PROPN
ma-117	231	12	θ2	θ2	ADP
ma-117	231	13	3∆t	3∆t	NUM
ma-117	231	14	1	1	NUM
ma-117	231	15	+	+	NUM
ma-117	231	16	θ2∆t	θ2∆t	NOUN
ma-117	231	17	∣∣∣∣2	∣∣∣∣2	NOUN
ma-117	231	18	=	=	PROPN
ma-117	231	19	θ2	θ2	ADP
ma-117	231	20	1(∆t)2	1(∆t)2	NUM
ma-117	231	21	(	(	PUNCT
ma-117	231	22	1	1	NUM
ma-117	231	23	+	+	NUM
ma-117	231	24	θ2∆t)2	θ2∆t)2	X
ma-117	232	1	+	+	CCONJ
ma-117	232	2	1	1	NUM
ma-117	232	3	(	(	PUNCT
ma-117	232	4	1	1	NUM
ma-117	232	5	+	+	NUM
ma-117	232	6	θ2∆t)2e	θ2∆t)2e	NOUN
ma-117	232	7	(	(	PUNCT
ma-117	232	8	|xk	|xk	X
ma-117	232	9	|2	|2	NUM
ma-117	232	10	)	)	PUNCT
ma-117	233	1	+	+	CCONJ
ma-117	233	2	θ2	θ2	PROPN
ma-117	233	3	3∆t	3∆t	NUM
ma-117	233	4	(	(	PUNCT
ma-117	233	5	1	1	NUM
ma-117	233	6	+	+	NUM
ma-117	233	7	θ2∆t)2	θ2∆t)2	X
ma-117	234	1	=	=	PROPN
ma-117	234	2	θ2	θ2	ADP
ma-117	234	3	1(∆t)2	1(∆t)2	NUM
ma-117	234	4	+	+	CCONJ
ma-117	234	5	θ2	θ2	PROPN
ma-117	234	6	3∆t	3∆t	NUM
ma-117	234	7	(	(	PUNCT
ma-117	234	8	1	1	NUM
ma-117	234	9	+	+	NUM
ma-117	234	10	θ2∆t)2	θ2∆t)2	X
ma-117	235	1	+	+	CCONJ
ma-117	235	2	1	1	NUM
ma-117	235	3	(	(	PUNCT
ma-117	235	4	1	1	NUM
ma-117	235	5	+	+	NUM
ma-117	235	6	θ2∆t)2e	θ2∆t)2e	NOUN
ma-117	235	7	(	(	PUNCT
ma-117	235	8	|xk	|xk	X
ma-117	235	9	|2	|2	NUM
ma-117	235	10	)	)	PUNCT
ma-117	235	11	=	=	SYM
ma-117	236	1	(	(	PUNCT
ma-117	236	2	θ2	θ2	ADV
ma-117	236	3	3	3	NUM
ma-117	236	4	+	+	CCONJ
ma-117	236	5	θ2	θ2	PROPN
ma-117	236	6	1∆t	1∆t	NUM
ma-117	236	7	)	)	PUNCT
ma-117	236	8	∆t	∆t	PROPN
ma-117	236	9	(	(	PUNCT
ma-117	236	10	1	1	NUM
ma-117	236	11	+	+	NUM
ma-117	236	12	θ2∆t)2	θ2∆t)2	X
ma-117	237	1	+	+	CCONJ
ma-117	237	2	(	(	PUNCT
ma-117	237	3	1	1	NUM
ma-117	237	4	1	1	NUM
ma-117	237	5	+	+	NUM
ma-117	237	6	θ2∆t	θ2∆t	NOUN
ma-117	237	7	)	)	PUNCT
ma-117	237	8	2	2	NUM
ma-117	237	9	{	{	PUNCT
ma-117	237	10	(	(	PUNCT
ma-117	237	11	1	1	NUM
ma-117	237	12	1	1	NUM
ma-117	237	13	+	+	NUM
ma-117	237	14	θ2∆t	θ2∆t	NOUN
ma-117	237	15	)	)	PUNCT
ma-117	237	16	2	2	NUM
ma-117	237	17	e	e	X
ma-117	237	18	(	(	PUNCT
ma-117	237	19	|xk−1|)2	|xk−1|)2	X
ma-117	237	20	+	+	CCONJ
ma-117	237	21	(	(	PUNCT
ma-117	237	22	θ2	θ2	ADV
ma-117	237	23	3	3	NUM
ma-117	237	24	+	+	CCONJ
ma-117	237	25	θ2	θ2	PROPN
ma-117	237	26	1∆t	1∆t	NUM
ma-117	237	27	)	)	PUNCT
ma-117	237	28	∆t	∆t	PROPN
ma-117	237	29	(	(	PUNCT
ma-117	237	30	1	1	NUM
ma-117	237	31	+	+	NUM
ma-117	237	32	θ2∆t)2	θ2∆t)2	X
ma-117	237	33	}	}	PUNCT
ma-117	237	34	=	=	SYM
ma-117	237	35	(	(	PUNCT
ma-117	237	36	1	1	NUM
ma-117	237	37	1	1	NUM
ma-117	237	38	+	+	NUM
ma-117	237	39	θ2∆t	θ2∆t	NOUN
ma-117	237	40	)	)	PUNCT
ma-117	237	41	4	4	NUM
ma-117	237	42	e	e	NOUN
ma-117	237	43	(	(	PUNCT
ma-117	237	44	|xk−1|)2	|xk−1|)2	X
ma-117	237	45	+	+	CCONJ
ma-117	237	46	(	(	PUNCT
ma-117	237	47	θ2	θ2	ADV
ma-117	237	48	3	3	NUM
ma-117	237	49	+	+	CCONJ
ma-117	237	50	θ2	θ2	PROPN
ma-117	237	51	1∆t	1∆t	NUM
ma-117	237	52	)	)	PUNCT
ma-117	237	53	∆t	∆t	VERB
ma-117	238	1	[	[	X
ma-117	238	2	(	(	PUNCT
ma-117	238	3	1	1	NUM
ma-117	238	4	1	1	NUM
ma-117	238	5	+	+	NUM
ma-117	238	6	θ2∆t	θ2∆t	NOUN
ma-117	238	7	)	)	PUNCT
ma-117	238	8	4	4	NUM
ma-117	239	1	+	+	CCONJ
ma-117	239	2	(	(	PUNCT
ma-117	239	3	1	1	NUM
ma-117	239	4	1	1	NUM
ma-117	239	5	+	+	NUM
ma-117	239	6	θ2∆t	θ2∆t	NOUN
ma-117	239	7	)	)	PUNCT
ma-117	239	8	2	2	NUM
ma-117	239	9	]	]	PUNCT
ma-117	239	10	...	...	PUNCT
ma-117	240	1	=	=	PUNCT
ma-117	240	2	(	(	PUNCT
ma-117	240	3	1	1	NUM
ma-117	240	4	1	1	NUM
ma-117	240	5	+	+	NUM
ma-117	240	6	θ2∆t	θ2∆t	NOUN
ma-117	240	7	)	)	PUNCT
ma-117	240	8	2(k+1	2(k+1	NUM
ma-117	240	9	)	)	PUNCT
ma-117	240	10	e	e	NOUN
ma-117	240	11	(	(	PUNCT
ma-117	240	12	|x0|)2	|x0|)2	NOUN
ma-117	240	13	+	+	CCONJ
ma-117	240	14	(	(	PUNCT
ma-117	240	15	θ2	θ2	ADV
ma-117	240	16	3	3	NUM
ma-117	240	17	+	+	CCONJ
ma-117	240	18	θ2	θ2	PROPN
ma-117	240	19	1∆t	1∆t	NUM
ma-117	240	20	)	)	PUNCT
ma-117	240	21	∆t	∆t	PROPN
ma-117	241	1	k+1∑	k+1∑	PROPN
ma-117	241	2	i=1	i=1	PROPN
ma-117	242	1	(	(	PUNCT
ma-117	242	2	1	1	NUM
ma-117	242	3	1	1	NUM
ma-117	242	4	+	+	NUM
ma-117	242	5	θ2∆t	θ2∆t	NOUN
ma-117	242	6	)	)	PUNCT
ma-117	242	7	2i	2i	NOUN
ma-117	242	8	by	by	ADP
ma-117	242	9	using	use	VERB
ma-117	242	10	the	the	DET
ma-117	242	11	geometrical	geometrical	ADJ
ma-117	242	12	sequence	sequence	NOUN
ma-117	242	13	and	and	CCONJ
ma-117	242	14	geometrical	geometrical	ADJ
ma-117	242	15	series	series	NOUN
ma-117	242	16	,	,	PUNCT
ma-117	242	17	we	we	PRON
ma-117	242	18	get	get	VERB
ma-117	242	19	:	:	PUNCT
ma-117	242	20	e	e	NOUN
ma-117	242	21	(	(	PUNCT
ma-117	242	22	∣∣xiemk+1	∣∣xiemk+1	VERB
ma-117	242	23	∣∣2	∣∣2	NUM
ma-117	242	24	)	)	PUNCT
ma-117	242	25	=	=	PUNCT
ma-117	243	1	(	(	PUNCT
ma-117	243	2	1	1	NUM
ma-117	243	3	1	1	NUM
ma-117	243	4	+	+	NUM
ma-117	243	5	θ2∆t	θ2∆t	NOUN
ma-117	243	6	)	)	PUNCT
ma-117	243	7	2(k+1	2(k+1	NUM
ma-117	243	8	)	)	PUNCT
ma-117	243	9	e	e	NOUN
ma-117	243	10	(	(	PUNCT
ma-117	243	11	|x0|)2	|x0|)2	NOUN
ma-117	243	12	+	+	CCONJ
ma-117	244	1	(	(	PUNCT
ma-117	244	2	θ2	θ2	ADV
ma-117	244	3	3	3	NUM
ma-117	244	4	+	+	CCONJ
ma-117	244	5	θ2	θ2	PROPN
ma-117	244	6	1∆t	1∆t	NUM
ma-117	244	7	)	)	PUNCT
ma-117	244	8	∆t	∆t	PROPN
ma-117	245	1	k+1∑	k+1∑	PROPN
ma-117	245	2	i=1	i=1	PROPN
ma-117	246	1	(	(	PUNCT
ma-117	246	2	1	1	NUM
ma-117	246	3	1	1	NUM
ma-117	246	4	+	+	NUM
ma-117	246	5	θ2∆t	θ2∆t	NOUN
ma-117	246	6	)	)	PUNCT
ma-117	246	7	2i	2i	NOUN
ma-117	246	8	we	we	PRON
ma-117	246	9	have	have	VERB
ma-117	246	10	the	the	DET
ma-117	246	11	geometric	geometric	ADJ
ma-117	246	12	sequence	sequence	NOUN
ma-117	246	13	and	and	CCONJ
ma-117	246	14	series	series	NOUN
ma-117	246	15	,	,	PUNCT
ma-117	246	16	converging	converge	VERB
ma-117	246	17	when	when	SCONJ
ma-117	246	18	|1	|1	PRON
ma-117	247	1	+	+	CCONJ
ma-117	247	2	θ2∆t|	θ2∆t|	ADP
ma-117	247	3	>	>	SYM
ma-117	247	4	1	1	NUM
ma-117	247	5	,	,	PUNCT
ma-117	247	6	by	by	ADP
ma-117	247	7	calculating	calculate	VERB
ma-117	247	8	limitof	limitof	PROPN
ma-117	247	9	the	the	DET
ma-117	247	10	equation	equation	NOUN
ma-117	247	11	below	below	ADV
ma-117	247	12	,	,	PUNCT
ma-117	247	13	for	for	ADP
ma-117	247	14	∆t	∆t	PROPN
ma-117	247	15	→	→	SYM
ma-117	247	16	0	0	NUM
ma-117	247	17	and	and	CCONJ
ma-117	247	18	k	k	PROPN
ma-117	247	19	→	→	SYM
ma-117	247	20	+	+	NOUN
ma-117	247	21	∞	∞	PROPN
ma-117	247	22	,	,	PUNCT
ma-117	247	23	we	we	PRON
ma-117	247	24	get	get	VERB
ma-117	247	25	:	:	PUNCT
ma-117	247	26	lim	lim	PROPN
ma-117	247	27	∆t→0	∆t→0	PROPN
ma-117	247	28	(	(	PUNCT
ma-117	247	29	lim	lim	PROPN
ma-117	247	30	k→+∞	k→+∞	PROPN
ma-117	247	31	e	e	PROPN
ma-117	247	32	[	[	PUNCT
ma-117	247	33	xiemk+1	xiemk+1	X
ma-117	247	34	]	]	X
ma-117	247	35	)	)	PUNCT
ma-117	248	1	=	=	SYM
ma-117	248	2	θ2	θ2	ADP
ma-117	248	3	3	3	NUM
ma-117	248	4	2θ2	2θ2	NUM
ma-117	248	5	https://doi.org/10.28924/ada/ma.3.8	https://doi.org/10.28924/ada/ma.3.8	PROPN
ma-117	248	6	eur	eur	NOUN
ma-117	248	7	.	.	PUNCT
ma-117	249	1	j.	j.	PROPN
ma-117	249	2	math	math	PROPN
ma-117	249	3	.	.	PUNCT
ma-117	250	1	anal	anal	PROPN
ma-117	250	2	.	.	PUNCT
ma-117	251	1	10.28924	10.28924	NUM
ma-117	251	2	/	/	SYM
ma-117	251	3	ada	ada	PROPN
ma-117	251	4	/	/	SYM
ma-117	251	5	ma.3.8	ma.3.8	PROPN
ma-117	251	6	11	11	NUM
ma-117	251	7	�	�	PROPN
ma-117	251	8	4	4	NUM
ma-117	251	9	.	.	PUNCT
ma-117	252	1	numerical	numerical	ADJ
ma-117	252	2	stabilities	stability	NOUN
ma-117	252	3	of	of	ADP
ma-117	252	4	geometric	geometric	ADJ
ma-117	252	5	brownian	brownian	ADJ
ma-117	252	6	motion	motion	NOUN
ma-117	252	7	4.1	4.1	NUM
ma-117	252	8	.	.	PUNCT
ma-117	253	1	explicit	explicit	ADJ
ma-117	253	2	solution	solution	NOUN
ma-117	253	3	of	of	ADP
ma-117	253	4	the	the	DET
ma-117	253	5	model	model	NOUN
ma-117	253	6	.	.	PUNCT
ma-117	254	1	geometric	geometric	ADJ
ma-117	254	2	brownian	brownian	ADJ
ma-117	254	3	motion	motion	NOUN
ma-117	254	4	known	know	VERB
ma-117	254	5	as	as	ADP
ma-117	254	6	exponential	exponential	ADJ
ma-117	254	7	brownianmotion	brownianmotion	NOUN
ma-117	254	8	is	be	AUX
ma-117	254	9	a	a	DET
ma-117	254	10	continuous	continuous	ADJ
ma-117	254	11	stochastic	stochastic	NOUN
ma-117	254	12	process	process	NOUN
ma-117	254	13	whose	whose	DET
ma-117	254	14	logarithm	logarithm	NOUN
ma-117	254	15	follows	follow	VERB
ma-117	254	16	a	a	DET
ma-117	254	17	brownian	brownian	ADJ
ma-117	254	18	motion	motion	NOUN
ma-117	254	19	.	.	PUNCT
ma-117	255	1	it	it	PRON
ma-117	255	2	is	be	AUX
ma-117	255	3	appliedin	appliedin	VERB
ma-117	255	4	the	the	DET
ma-117	255	5	mathematical	mathematical	ADJ
ma-117	255	6	modeling	modeling	NOUN
ma-117	255	7	of	of	ADP
ma-117	255	8	certain	certain	ADJ
ma-117	255	9	courses	course	NOUN
ma-117	255	10	in	in	ADP
ma-117	255	11	the	the	DET
ma-117	255	12	financial	financial	ADJ
ma-117	255	13	markets	market	NOUN
ma-117	255	14	[	[	X
ma-117	255	15	26	26	NUM
ma-117	255	16	]	]	PUNCT
ma-117	255	17	.	.	PUNCT
ma-117	256	1	it	it	PRON
ma-117	256	2	represents	represent	VERB
ma-117	256	3	areasonable	areasonable	ADJ
ma-117	256	4	approximation	approximation	NOUN
ma-117	256	5	of	of	ADP
ma-117	256	6	the	the	DET
ma-117	256	7	evolution	evolution	NOUN
ma-117	256	8	of	of	ADP
ma-117	256	9	stock	stock	NOUN
ma-117	256	10	market	market	NOUN
ma-117	256	11	prices	price	NOUN
ma-117	256	12	,	,	PUNCT
ma-117	256	13	because	because	SCONJ
ma-117	256	14	a	a	DET
ma-117	256	15	quantity	quantity	NOUN
ma-117	256	16	which	which	PRON
ma-117	256	17	followsa	followsa	VERB
ma-117	256	18	geometric	geometric	ADJ
ma-117	256	19	brownian	brownian	ADJ
ma-117	256	20	motion	motion	NOUN
ma-117	256	21	takes	take	VERB
ma-117	256	22	all	all	DET
ma-117	256	23	strictly	strictly	ADV
ma-117	256	24	positive	positive	ADJ
ma-117	256	25	values	value	NOUN
ma-117	256	26	and	and	CCONJ
ma-117	256	27	only	only	ADV
ma-117	256	28	the	the	DET
ma-117	256	29	elementary	elementary	ADJ
ma-117	256	30	changesundergone	changesundergone	NOUN
ma-117	256	31	by	by	ADP
ma-117	256	32	the	the	DET
ma-117	256	33	random	random	ADJ
ma-117	256	34	variable	variable	NOUN
ma-117	256	35	are	be	AUX
ma-117	256	36	significant	significant	ADJ
ma-117	256	37	.	.	PUNCT
ma-117	257	1	the	the	DET
ma-117	257	2	geometric	geometric	ADJ
ma-117	257	3	brownian	brownian	ADJ
ma-117	257	4	motion	motion	NOUN
ma-117	257	5	xt	xt	PROPN
ma-117	257	6	is	be	AUX
ma-117	257	7	a	a	DET
ma-117	257	8	process	process	NOUN
ma-117	257	9	which	which	PRON
ma-117	257	10	is	be	AUX
ma-117	257	11	written	write	VERB
ma-117	257	12	in	in	ADP
ma-117	257	13	the	the	DET
ma-117	257	14	form	form	NOUN
ma-117	257	15	[	[	X
ma-117	257	16	13	13	NUM
ma-117	257	17	]	]	PUNCT
ma-117	257	18	:	:	PUNCT
ma-117	257	19	{	{	PUNCT
ma-117	257	20	dxt	dxt	X
ma-117	257	21	=	=	PUNCT
ma-117	257	22	θ1xtdt	θ1xtdt	PROPN
ma-117	257	23	+	+	CCONJ
ma-117	257	24	θ2xtdbt	θ2xtdbt	PROPN
ma-117	257	25	x(0	x(0	NUM
ma-117	257	26	)	)	PUNCT
ma-117	258	1	=	=	PUNCT
ma-117	259	1	x0	x0	PROPN
ma-117	259	2	∀θ1	∀θ1	NOUN
ma-117	259	3	,	,	PUNCT
ma-117	259	4	θ2	θ2	PROPN
ma-117	259	5	∈	∈	PROPN
ma-117	259	6	r	r	NOUN
ma-117	259	7	(	(	PUNCT
ma-117	259	8	4.1	4.1	NUM
ma-117	259	9	)	)	PUNCT
ma-117	259	10	this	this	DET
ma-117	259	11	process	process	NOUN
ma-117	259	12	admits	admit	VERB
ma-117	259	13	as	as	ADP
ma-117	259	14	an	an	DET
ma-117	259	15	explicit	explicit	ADJ
ma-117	259	16	solution	solution	NOUN
ma-117	259	17	:	:	PUNCT
ma-117	259	18	xt	xt	NOUN
ma-117	260	1	=	=	PUNCT
ma-117	260	2	x0e	x0e	PUNCT
ma-117	260	3	{	{	PUNCT
ma-117	260	4	(	(	PUNCT
ma-117	260	5	θ1−	θ1−	VERB
ma-117	260	6	1	1	NUM
ma-117	260	7	2	2	NUM
ma-117	260	8	θ2	θ2	ADP
ma-117	260	9	2)t+θ2bt	2)t+θ2bt	NUM
ma-117	260	10	}	}	PUNCT
ma-117	260	11	(	(	PUNCT
ma-117	260	12	4.2	4.2	NUM
ma-117	260	13	)	)	PUNCT
ma-117	260	14	the	the	DET
ma-117	260	15	variable	variable	NOUN
ma-117	260	16	on	on	ADP
ma-117	260	17	the	the	DET
ma-117	260	18	right	right	ADJ
ma-117	260	19	hand	hand	NOUN
ma-117	260	20	follows	follow	VERB
ma-117	260	21	a	a	DET
ma-117	260	22	normal	normal	ADJ
ma-117	260	23	distribution	distribution	NOUN
ma-117	260	24	,	,	PUNCT
ma-117	260	25	it	it	PRON
ma-117	260	26	can	can	AUX
ma-117	260	27	also	also	ADV
ma-117	260	28	be	be	AUX
ma-117	260	29	written	write	VERB
ma-117	260	30	in	in	ADP
ma-117	260	31	the	the	DET
ma-117	260	32	form	form	NOUN
ma-117	260	33	:	:	PUNCT
ma-117	260	34	xt	xt	PROPN
ma-117	260	35	=	=	PROPN
ma-117	260	36	xse	xse	PROPN
ma-117	260	37	{	{	PUNCT
ma-117	260	38	(	(	PUNCT
ma-117	260	39	θ1−	θ1−	NUM
ma-117	260	40	1	1	NUM
ma-117	260	41	2	2	NUM
ma-117	260	42	θ2	θ2	ADP
ma-117	260	43	2)t+θ2(bt−bs	2)t+θ2(bt−b	NOUN
ma-117	260	44	)	)	PUNCT
ma-117	260	45	}	}	PUNCT
ma-117	260	46	(	(	PUNCT
ma-117	260	47	4.3	4.3	NUM
ma-117	260	48	)	)	PUNCT
ma-117	260	49	the	the	DET
ma-117	260	50	conditionnal	conditionnal	ADJ
ma-117	260	51	mean	mean	NOUN
ma-117	260	52	is	be	AUX
ma-117	260	53	:	:	PUNCT
ma-117	260	54	e	e	X
ma-117	260	55	(	(	PUNCT
ma-117	260	56	xt	xt	ADP
ma-117	260	57	|xs	|xs	NUM
ma-117	260	58	)	)	PUNCT
ma-117	260	59	=	=	SYM
ma-117	260	60	xse	xse	PROPN
ma-117	260	61	θ1(t−s	θ1(t−s	PROPN
ma-117	260	62	)	)	PUNCT
ma-117	260	63	(	(	PUNCT
ma-117	260	64	4.4)the	4.4)the	DET
ma-117	260	65	(	(	PUNCT
ma-117	260	66	4.3	4.3	NUM
ma-117	260	67	)	)	PUNCT
ma-117	260	68	process	process	NOUN
ma-117	260	69	is	be	AUX
ma-117	260	70	often	often	ADV
ma-117	260	71	widely	widely	ADV
ma-117	260	72	used	use	VERB
ma-117	260	73	to	to	PART
ma-117	260	74	model	model	VERB
ma-117	260	75	the	the	DET
ma-117	260	76	price	price	NOUN
ma-117	260	77	of	of	ADP
ma-117	260	78	a	a	DET
ma-117	260	79	financial	financial	ADJ
ma-117	260	80	asset	asset	NOUN
ma-117	260	81	the	the	DET
ma-117	260	82	return	return	NOUN
ma-117	260	83	on	on	ADP
ma-117	260	84	theasset	theasset	NOUN
ma-117	260	85	between	between	ADP
ma-117	260	86	two	two	NUM
ma-117	260	87	dates	date	NOUN
ma-117	260	88	is	be	AUX
ma-117	260	89	measured	measure	VERB
ma-117	260	90	by	by	ADP
ma-117	260	91	the	the	DET
ma-117	260	92	difference	difference	NOUN
ma-117	260	93	in	in	ADP
ma-117	260	94	the	the	DET
ma-117	260	95	logarithms	logarithm	NOUN
ma-117	260	96	of	of	ADP
ma-117	260	97	the	the	DET
ma-117	260	98	prices	price	NOUN
ma-117	260	99	and	and	CCONJ
ma-117	260	100	is	be	AUX
ma-117	260	101	givenby	givenby	VERB
ma-117	260	102	the	the	DET
ma-117	260	103	gaussian	gaussian	ADJ
ma-117	260	104	variable	variable	NOUN
ma-117	260	105	below	below	ADV
ma-117	260	106	:	:	PUNCT
ma-117	260	107	{	{	PUNCT
ma-117	260	108	θ1	θ1	NOUN
ma-117	260	109	−	−	PROPN
ma-117	260	110	1	1	NUM
ma-117	260	111	2	2	NUM
ma-117	260	112	θ2	θ2	ADP
ma-117	260	113	2	2	NUM
ma-117	260	114	}	}	PUNCT
ma-117	260	115	(	(	PUNCT
ma-117	260	116	t	t	PROPN
ma-117	260	117	−	−	PROPN
ma-117	260	118	s	s	PART
ma-117	260	119	)	)	PUNCT
ma-117	260	120	+	+	CCONJ
ma-117	260	121	θ2	θ2	PROPN
ma-117	260	122	(	(	PUNCT
ma-117	260	123	bt	bt	INTJ
ma-117	260	124	−	−	PROPN
ma-117	260	125	bs	bs	PROPN
ma-117	260	126	)	)	PUNCT
ma-117	260	127	the	the	DET
ma-117	260	128	mean	mean	NOUN
ma-117	260	129	and	and	CCONJ
ma-117	260	130	the	the	DET
ma-117	260	131	mean	mean	ADJ
ma-117	260	132	-	-	PUNCT
ma-117	260	133	square	square	NOUN
ma-117	260	134	give	give	VERB
ma-117	260	135	respectively	respectively	ADV
ma-117	260	136	:	:	PUNCT
ma-117	260	137	e(xt	e(xt	NOUN
ma-117	260	138	)	)	PUNCT
ma-117	260	139	=	=	SYM
ma-117	260	140	x0e	x0e	PUNCT
ma-117	260	141	θ1	θ1	PROPN
ma-117	260	142	t	t	PROPN
ma-117	260	143	e(x2	e(x2	NOUN
ma-117	260	144	t	t	PROPN
ma-117	260	145	)	)	PUNCT
ma-117	261	1	=	=	SYM
ma-117	262	1	x2	x2	NUM
ma-117	262	2	0e	0e	NOUN
ma-117	262	3	(	(	PUNCT
ma-117	262	4	2θ1+θ2	2θ1+θ2	NUM
ma-117	262	5	2)t	2)t	NUM
ma-117	262	6	(	(	PUNCT
ma-117	262	7	4.5	4.5	NUM
ma-117	262	8	)	)	PUNCT
ma-117	262	9	remark	remark	NOUN
ma-117	262	10	4.1	4.1	NUM
ma-117	262	11	.	.	PUNCT
ma-117	263	1	it	it	PRON
ma-117	263	2	should	should	AUX
ma-117	263	3	be	be	AUX
ma-117	263	4	noted	note	VERB
ma-117	263	5	that:(1	that:(1	PROPN
ma-117	263	6	)	)	PUNCT
ma-117	263	7	for	for	ADP
ma-117	263	8	mean	mean	NOUN
ma-117	263	9	if	if	SCONJ
ma-117	263	10	t	t	PROPN
ma-117	263	11	→∞	→∞	X
ma-117	263	12	and	and	CCONJ
ma-117	263	13	θ1	θ1	NOUN
ma-117	263	14	<	<	X
ma-117	263	15	0	0	NUM
ma-117	263	16	we	we	PRON
ma-117	263	17	have	have	VERB
ma-117	263	18	:	:	PUNCT
ma-117	263	19	lim	lim	PROPN
ma-117	263	20	t→∞	t→∞	PRON
ma-117	263	21	e(xt	e(xt	NOUN
ma-117	263	22	)	)	PUNCT
ma-117	264	1	=	=	SYM
ma-117	264	2	lim	lim	PROPN
ma-117	264	3	k→∞	k→∞	NOUN
ma-117	264	4	x0e	x0e	PUNCT
ma-117	265	1	θ1	θ1	NOUN
ma-117	265	2	t	t	NOUN
ma-117	265	3	=	=	SYM
ma-117	265	4	0	0	NUM
ma-117	265	5	(	(	PUNCT
ma-117	265	6	2	2	NUM
ma-117	265	7	)	)	PUNCT
ma-117	265	8	for	for	ADP
ma-117	265	9	mean	mean	ADJ
ma-117	265	10	-	-	PUNCT
ma-117	265	11	square	square	NOUN
ma-117	265	12	if	if	SCONJ
ma-117	265	13	(	(	PUNCT
ma-117	265	14	2θ1	2θ1	NUM
ma-117	265	15	+	+	NUM
ma-117	265	16	θ2	θ2	PROPN
ma-117	265	17	2	2	NUM
ma-117	265	18	)	)	PUNCT
ma-117	265	19	<	<	X
ma-117	265	20	0	0	PUNCT
ma-117	265	21	and	and	CCONJ
ma-117	265	22	t	t	PROPN
ma-117	265	23	→∞	→∞	PROPN
ma-117	266	1	i.e	i.e	PROPN
ma-117	266	2	lim	lim	PROPN
ma-117	266	3	t→∞	t→∞	PRON
ma-117	266	4	e(x2	e(x2	NOUN
ma-117	266	5	t	t	NOUN
ma-117	266	6	)	)	PUNCT
ma-117	267	1	=	=	SYM
ma-117	267	2	0	0	NUM
ma-117	267	3	with	with	ADP
ma-117	267	4	(	(	PUNCT
ma-117	267	5	2θ1	2θ1	NUM
ma-117	267	6	+	+	NUM
ma-117	267	7	θ2	θ2	PROPN
ma-117	267	8	2	2	NUM
ma-117	267	9	)	)	PUNCT
ma-117	267	10	<	<	X
ma-117	267	11	0	0	X
ma-117	267	12	.	.	PUNCT
ma-117	267	13	now	now	ADV
ma-117	267	14	,	,	PUNCT
ma-117	267	15	let	let	VERB
ma-117	267	16	’s	’s	PRON
ma-117	267	17	analyze	analyze	VERB
ma-117	267	18	the	the	DET
ma-117	267	19	stabilities	stability	NOUN
ma-117	267	20	of	of	ADP
ma-117	267	21	some	some	DET
ma-117	267	22	numerical	numerical	ADJ
ma-117	267	23	schemes	scheme	NOUN
ma-117	267	24	(	(	PUNCT
ma-117	267	25	euler	euler	NOUN
ma-117	267	26	-	-	PUNCT
ma-117	267	27	maruyama	maruyama	NOUN
ma-117	267	28	,	,	PUNCT
ma-117	267	29	milshtein	milshtein	PROPN
ma-117	267	30	andimplicit	andimplicit	PROPN
ma-117	267	31	euler	euler	PROPN
ma-117	267	32	-	-	PUNCT
ma-117	267	33	maruyama	maruyama	NOUN
ma-117	267	34	)	)	PUNCT
ma-117	267	35	in	in	ADP
ma-117	267	36	mean	mean	ADJ
ma-117	267	37	and	and	CCONJ
ma-117	267	38	mean	mean	ADJ
ma-117	267	39	-	-	PUNCT
ma-117	267	40	square	square	NOUN
ma-117	267	41	.	.	PUNCT
ma-117	268	1	https://doi.org/10.28924/ada/ma.3.8	https://doi.org/10.28924/ada/ma.3.8	NUM
ma-117	268	2	eur	eur	PROPN
ma-117	268	3	.	.	PUNCT
ma-117	269	1	j.	j.	PROPN
ma-117	269	2	math	math	PROPN
ma-117	269	3	.	.	PUNCT
ma-117	270	1	anal	anal	PROPN
ma-117	270	2	.	.	PUNCT
ma-117	271	1	10.28924	10.28924	NUM
ma-117	271	2	/	/	SYM
ma-117	271	3	ada	ada	PROPN
ma-117	271	4	/	/	SYM
ma-117	271	5	ma.3.8	ma.3.8	PROPN
ma-117	271	6	124.2	124.2	NUM
ma-117	271	7	.	.	PUNCT
ma-117	271	8	euler	euler	PROPN
ma-117	271	9	-	-	PUNCT
ma-117	271	10	maruyama	maruyama	PROPN
ma-117	271	11	scheme	scheme	NOUN
ma-117	271	12	stabilities	stability	NOUN
ma-117	271	13	.	.	PUNCT
ma-117	272	1	the	the	DET
ma-117	272	2	euler	euler	NOUN
ma-117	272	3	-	-	PUNCT
ma-117	272	4	maruyama	maruyama	NOUN
ma-117	272	5	scheme	scheme	NOUN
ma-117	272	6	associated	associate	VERB
ma-117	272	7	to	to	ADP
ma-117	272	8	(	(	PUNCT
ma-117	272	9	4.1	4.1	NUM
ma-117	272	10	)	)	PUNCT
ma-117	272	11	is	be	AUX
ma-117	272	12	:	:	PUNCT
ma-117	272	13	xemk+1	xemk+1	PROPN
ma-117	272	14	=	=	SYM
ma-117	272	15	xk	xk	PROPN
ma-117	273	1	+	+	CCONJ
ma-117	273	2	θ1xk∆t	θ1xk∆t	NOUN
ma-117	273	3	+	+	CCONJ
ma-117	273	4	θ2xk∆bk	θ2xk∆bk	ADV
ma-117	273	5	xemk+1	xemk+1	PUNCT
ma-117	273	6	=	=	SYM
ma-117	273	7	xk	xk	X
ma-117	273	8	(	(	PUNCT
ma-117	273	9	1	1	NUM
ma-117	273	10	+	+	NUM
ma-117	273	11	θ1∆t	θ1∆t	NOUN
ma-117	273	12	)	)	PUNCT
ma-117	274	1	+	+	NUM
ma-117	274	2	θ2xk	θ2xk	PUNCT
ma-117	274	3	√	√	ADP
ma-117	274	4	∆tzk	∆tzk	NOUN
ma-117	274	5	(	(	PUNCT
ma-117	274	6	4.6	4.6	NUM
ma-117	274	7	)	)	PUNCT
ma-117	274	8	4.2.1	4.2.1	NOUN
ma-117	274	9	.	.	PUNCT
ma-117	275	1	mean	mean	VERB
ma-117	275	2	stability	stability	NOUN
ma-117	275	3	of	of	ADP
ma-117	275	4	euler	euler	NOUN
ma-117	275	5	-	-	PUNCT
ma-117	275	6	maruyama	maruyama	NOUN
ma-117	275	7	scheme	scheme	NOUN
ma-117	275	8	.	.	PUNCT
ma-117	276	1	theorem	theorem	VERB
ma-117	276	2	4.1	4.1	NUM
ma-117	276	3	.	.	PUNCT
ma-117	277	1	(	(	PUNCT
ma-117	277	2	mean	mean	VERB
ma-117	277	3	stability	stability	NOUN
ma-117	277	4	of	of	ADP
ma-117	277	5	euler	euler	NOUN
ma-117	277	6	-	-	PUNCT
ma-117	277	7	maruyama	maruyama	NOUN
ma-117	277	8	scheme	scheme	NOUN
ma-117	277	9	)	)	PUNCT
ma-117	277	10	the	the	DET
ma-117	277	11	euler	euler	NOUN
ma-117	277	12	-	-	PUNCT
ma-117	277	13	maruyama	maruyama	NOUN
ma-117	277	14	scheme	scheme	NOUN
ma-117	277	15	(	(	PUNCT
ma-117	277	16	4.6	4.6	NUM
ma-117	277	17	)	)	PUNCT
ma-117	277	18	associated	associate	VERB
ma-117	277	19	to	to	ADP
ma-117	277	20	(	(	PUNCT
ma-117	277	21	4.1	4.1	NUM
ma-117	277	22	)	)	PUNCT
ma-117	277	23	model	model	NOUN
ma-117	277	24	is	be	AUX
ma-117	277	25	mean	mean	ADV
ma-117	277	26	asymptotically	asymptotically	ADV
ma-117	277	27	stable	stable	ADJ
ma-117	277	28	if	if	SCONJ
ma-117	277	29	e	e	X
ma-117	277	30	[	[	PUNCT
ma-117	277	31	xemk+1	xemk+1	X
ma-117	277	32	]	]	PUNCT
ma-117	277	33	=	=	SYM
ma-117	277	34	(	(	PUNCT
ma-117	277	35	1	1	NUM
ma-117	277	36	+	+	X
ma-117	277	37	θ2∆t)k+1e	θ2∆t)k+1e	NUM
ma-117	277	38	(	(	PUNCT
ma-117	277	39	x0	x0	PROPN
ma-117	277	40	)	)	PUNCT
ma-117	277	41	with	with	ADP
ma-117	277	42	|1	|1	PRON
ma-117	278	1	+	+	NUM
ma-117	278	2	θ1∆t|	θ1∆t|	PROPN
ma-117	278	3	<	<	X
ma-117	278	4	1	1	NUM
ma-117	278	5	and	and	CCONJ
ma-117	278	6	lim	lim	PROPN
ma-117	278	7	∆t→0	∆t→0	PROPN
ma-117	278	8	(	(	PUNCT
ma-117	278	9	lim	lim	PROPN
ma-117	278	10	k→+∞	k→+∞	PROPN
ma-117	278	11	e	e	PROPN
ma-117	278	12	[	[	PUNCT
ma-117	278	13	xemk+1	xemk+1	X
ma-117	278	14	]	]	X
ma-117	278	15	)	)	PUNCT
ma-117	278	16	=	=	SYM
ma-117	278	17	0	0	NUM
ma-117	278	18	proof	proof	NOUN
ma-117	278	19	.	.	PUNCT
ma-117	279	1	by	by	ADP
ma-117	279	2	calculating	calculate	VERB
ma-117	279	3	the	the	DET
ma-117	279	4	mean	mean	NOUN
ma-117	279	5	of	of	ADP
ma-117	279	6	the	the	DET
ma-117	279	7	expression(4.6	expression(4.6	NOUN
ma-117	279	8	)	)	PUNCT
ma-117	279	9	,	,	PUNCT
ma-117	279	10	we	we	PRON
ma-117	279	11	obtain	obtain	VERB
ma-117	279	12	:	:	PUNCT
ma-117	279	13	e	e	X
ma-117	279	14	[	[	PUNCT
ma-117	279	15	xemk+1	xemk+1	X
ma-117	279	16	]	]	PUNCT
ma-117	279	17	=	=	SYM
ma-117	279	18	e	e	X
ma-117	279	19	[	[	PUNCT
ma-117	279	20	xt(1	xt(1	PROPN
ma-117	279	21	+	+	NUM
ma-117	279	22	θ1∆t	θ1∆t	NOUN
ma-117	279	23	)	)	PUNCT
ma-117	279	24	+	+	CCONJ
ma-117	279	25	θ2xt	θ2xt	ADP
ma-117	279	26	√	√	VERB
ma-117	279	27	∆tzt	∆tzt	NOUN
ma-117	279	28	]	]	PUNCT
ma-117	279	29	=	=	PUNCT
ma-117	279	30	e	e	X
ma-117	279	31	[	[	X
ma-117	279	32	xt(1	xt(1	PROPN
ma-117	279	33	+	+	NUM
ma-117	279	34	θ1∆t	θ1∆t	NOUN
ma-117	279	35	)	)	PUNCT
ma-117	279	36	]	]	PUNCT
ma-117	280	1	+	+	CCONJ
ma-117	280	2	e	e	X
ma-117	280	3	[	[	PUNCT
ma-117	280	4	θ2	θ2	ADV
ma-117	280	5	√	√	VERB
ma-117	280	6	∆txtzt	∆txtzt	NOUN
ma-117	280	7	]	]	PUNCT
ma-117	281	1	=	=	PUNCT
ma-117	281	2	e	e	X
ma-117	281	3	[	[	X
ma-117	281	4	(	(	PUNCT
ma-117	281	5	1	1	NUM
ma-117	281	6	+	+	NOUN
ma-117	281	7	θ1∆t)xt	θ1∆t)xt	PROPN
ma-117	281	8	]	]	PUNCT
ma-117	282	1	+	+	CCONJ
ma-117	282	2	e	e	X
ma-117	282	3	[	[	PUNCT
ma-117	282	4	θ2	θ2	ADV
ma-117	282	5	√	√	ADV
ma-117	282	6	∆t	∆t	PROPN
ma-117	282	7	]	]	PUNCT
ma-117	282	8	e	e	X
ma-117	283	1	[	[	X
ma-117	283	2	xt	xt	X
ma-117	283	3	]	]	X
ma-117	283	4	e	e	X
ma-117	284	1	[	[	X
ma-117	284	2	zt	zt	X
ma-117	284	3	]	]	PUNCT
ma-117	284	4	as	as	ADP
ma-117	284	5	zk	zk	PROPN
ma-117	284	6	'	'	PART
ma-117	284	7	n	n	CCONJ
ma-117	284	8	(	(	PUNCT
ma-117	284	9	0	0	NUM
ma-117	284	10	,	,	PUNCT
ma-117	284	11	1	1	NUM
ma-117	284	12	)	)	PUNCT
ma-117	284	13	e(zk	e(zk	PROPN
ma-117	284	14	)	)	PUNCT
ma-117	284	15	=	=	PUNCT
ma-117	285	1	0	0	NUM
ma-117	285	2	e	e	X
ma-117	285	3	[	[	X
ma-117	285	4	xk+1	xk+1	X
ma-117	285	5	]	]	X
ma-117	285	6	=	=	SYM
ma-117	285	7	(	(	PUNCT
ma-117	285	8	1	1	NUM
ma-117	285	9	+	+	NUM
ma-117	285	10	θ1∆t)e(xt	θ1∆t)e(xt	NOUN
ma-117	285	11	)	)	PUNCT
ma-117	285	12	=	=	SYM
ma-117	285	13	(	(	PUNCT
ma-117	285	14	1	1	NUM
ma-117	285	15	+	+	NUM
ma-117	285	16	θ1∆t	θ1∆t	NOUN
ma-117	285	17	)	)	PUNCT
ma-117	285	18	(	(	PUNCT
ma-117	285	19	(	(	PUNCT
ma-117	285	20	1	1	NUM
ma-117	285	21	+	+	NUM
ma-117	285	22	θ2∆t)e	θ2∆t)e	NOUN
ma-117	285	23	(	(	PUNCT
ma-117	285	24	xk−1	xk−1	PROPN
ma-117	285	25	)	)	PUNCT
ma-117	285	26	)	)	PUNCT
ma-117	286	1	=	=	PUNCT
ma-117	286	2	(	(	PUNCT
ma-117	286	3	1	1	NUM
ma-117	286	4	+	+	NUM
ma-117	286	5	θ2∆t)2e	θ2∆t)2e	NOUN
ma-117	286	6	(	(	PUNCT
ma-117	286	7	xk−1	xk−1	PROPN
ma-117	286	8	)	)	PUNCT
ma-117	286	9	=	=	PUNCT
ma-117	287	1	(	(	PUNCT
ma-117	287	2	1	1	NUM
ma-117	287	3	+	+	NUM
ma-117	287	4	θ1∆t)2	θ1∆t)2	NOUN
ma-117	287	5	(	(	PUNCT
ma-117	287	6	(	(	PUNCT
ma-117	287	7	1	1	NUM
ma-117	287	8	+	+	NUM
ma-117	287	9	θ2∆t)e	θ2∆t)e	NOUN
ma-117	287	10	(	(	PUNCT
ma-117	287	11	xk−2	xk−2	PROPN
ma-117	287	12	)	)	PUNCT
ma-117	287	13	)	)	PUNCT
ma-117	287	14	...	...	PUNCT
ma-117	288	1	=	=	PUNCT
ma-117	288	2	(	(	PUNCT
ma-117	288	3	1	1	NUM
ma-117	288	4	+	+	X
ma-117	288	5	θ2∆t)k+1e	θ2∆t)k+1e	NUM
ma-117	288	6	(	(	PUNCT
ma-117	288	7	x0	x0	PROPN
ma-117	288	8	)	)	PUNCT
ma-117	288	9	we	we	PRON
ma-117	288	10	get	get	VERB
ma-117	288	11	the	the	DET
ma-117	288	12	following	follow	VERB
ma-117	288	13	geometric	geometric	ADJ
ma-117	288	14	sequence	sequence	NOUN
ma-117	288	15	:	:	PUNCT
ma-117	288	16	e	e	X
ma-117	288	17	[	[	PUNCT
ma-117	288	18	xemk+1	xemk+1	X
ma-117	288	19	]	]	PUNCT
ma-117	288	20	=	=	SYM
ma-117	288	21	(	(	PUNCT
ma-117	288	22	1	1	NUM
ma-117	288	23	+	+	X
ma-117	288	24	θ2∆t)k+1e	θ2∆t)k+1e	NUM
ma-117	288	25	(	(	PUNCT
ma-117	288	26	x0	x0	PROPN
ma-117	288	27	)	)	PUNCT
ma-117	288	28	which	which	PRON
ma-117	288	29	converges	converge	VERB
ma-117	288	30	if	if	SCONJ
ma-117	288	31	|1	|1	PRON
ma-117	289	1	+	+	CCONJ
ma-117	289	2	θ2∆t|	θ2∆t|	PRON
ma-117	289	3	<	<	AUX
ma-117	289	4	1	1	NUM
ma-117	289	5	et	et	NOUN
ma-117	289	6	nd	nd	ADV
ma-117	289	7	passing	pass	VERB
ma-117	289	8	to	to	ADP
ma-117	289	9	the	the	DET
ma-117	289	10	limit	limit	NOUN
ma-117	289	11	for	for	ADP
ma-117	289	12	a	a	DET
ma-117	289	13	∆t	∆t	PROPN
ma-117	289	14	→	→	SYM
ma-117	289	15	0	0	NUM
ma-117	289	16	and	and	CCONJ
ma-117	289	17	k	k	PROPN
ma-117	289	18	→	→	SYM
ma-117	289	19	+	+	NOUN
ma-117	289	20	∞	∞	PROPN
ma-117	289	21	,	,	PUNCT
ma-117	289	22	we	we	PRON
ma-117	289	23	obtain	obtain	VERB
ma-117	289	24	:	:	PUNCT
ma-117	289	25	lim	lim	PROPN
ma-117	289	26	∆t→0	∆t→0	PROPN
ma-117	289	27	(	(	PUNCT
ma-117	289	28	lim	lim	PROPN
ma-117	289	29	k→+∞	k→+∞	PROPN
ma-117	289	30	e	e	PROPN
ma-117	289	31	[	[	PUNCT
ma-117	289	32	xemk+1	xemk+1	X
ma-117	289	33	]	]	X
ma-117	289	34	)	)	PUNCT
ma-117	289	35	=	=	SYM
ma-117	289	36	0	0	NUM
ma-117	289	37	�	�	PROPN
ma-117	289	38	https://doi.org/10.28924/ada/ma.3.8	https://doi.org/10.28924/ada/ma.3.8	PROPN
ma-117	289	39	eur	eur	PROPN
ma-117	289	40	.	.	PUNCT
ma-117	290	1	j.	j.	PROPN
ma-117	290	2	math	math	PROPN
ma-117	290	3	.	.	PUNCT
ma-117	291	1	anal	anal	PROPN
ma-117	291	2	.	.	PUNCT
ma-117	292	1	10.28924	10.28924	NUM
ma-117	292	2	/	/	SYM
ma-117	292	3	ada	ada	PROPN
ma-117	292	4	/	/	SYM
ma-117	292	5	ma.3.8	ma.3.8	PROPN
ma-117	292	6	134.2.2	134.2.2	NUM
ma-117	292	7	.	.	PUNCT
ma-117	293	1	mean	mean	ADJ
ma-117	293	2	-	-	PUNCT
ma-117	293	3	square	square	ADJ
ma-117	293	4	stability	stability	NOUN
ma-117	293	5	of	of	ADP
ma-117	293	6	euler	euler	NOUN
ma-117	293	7	-	-	PUNCT
ma-117	293	8	maruyama	maruyama	NOUN
ma-117	293	9	scheme	scheme	NOUN
ma-117	293	10	.	.	PUNCT
ma-117	294	1	theorem	theorem	VERB
ma-117	294	2	4.2	4.2	NUM
ma-117	294	3	.	.	PUNCT
ma-117	295	1	(	(	PUNCT
ma-117	295	2	mean	mean	ADJ
ma-117	295	3	-	-	PUNCT
ma-117	295	4	square	square	ADJ
ma-117	295	5	stability	stability	NOUN
ma-117	295	6	of	of	ADP
ma-117	295	7	euler	euler	NOUN
ma-117	295	8	-	-	PUNCT
ma-117	295	9	maruyama	maruyama	NOUN
ma-117	295	10	scheme	scheme	NOUN
ma-117	295	11	)	)	PUNCT
ma-117	295	12	the	the	DET
ma-117	295	13	euler	euler	NOUN
ma-117	295	14	-	-	PUNCT
ma-117	295	15	maruyama	maruyama	NOUN
ma-117	295	16	scheme(4.6	scheme(4.6	NOUN
ma-117	295	17	)	)	PUNCT
ma-117	295	18	associated	associate	VERB
ma-117	295	19	to	to	ADP
ma-117	295	20	(	(	PUNCT
ma-117	295	21	4.1	4.1	NUM
ma-117	295	22	)	)	PUNCT
ma-117	295	23	model	model	NOUN
ma-117	295	24	is	be	AUX
ma-117	295	25	mean	mean	ADJ
ma-117	295	26	-	-	PUNCT
ma-117	295	27	square	square	ADJ
ma-117	295	28	asymptotically	asymptotically	ADV
ma-117	295	29	stable	stable	ADJ
ma-117	295	30	if	if	SCONJ
ma-117	295	31	:	:	PUNCT
ma-117	295	32	e	e	X
ma-117	295	33	(	(	PUNCT
ma-117	295	34	∣∣xemk+1	∣∣xemk+1	VERB
ma-117	295	35	∣∣2	∣∣2	NUM
ma-117	295	36	)	)	PUNCT
ma-117	295	37	=	=	NOUN
ma-117	296	1	(	(	PUNCT
ma-117	296	2	|(1	|(1	NOUN
ma-117	296	3	+	+	CCONJ
ma-117	296	4	θ1∆t)|2	θ1∆t)|2	NOUN
ma-117	296	5	+	+	CCONJ
ma-117	296	6	∣∣∣θ2	∣∣∣θ2	PROPN
ma-117	296	7	√	√	ADP
ma-117	296	8	∆t	∆t	PROPN
ma-117	296	9	∣∣∣2)2k+2	∣∣∣2)2k+2	PROPN
ma-117	296	10	e	e	X
ma-117	296	11	(	(	PUNCT
ma-117	296	12	|x0|2	|x0|2	PROPN
ma-117	296	13	)	)	PUNCT
ma-117	296	14	,	,	PUNCT
ma-117	296	15	with	with	ADP
ma-117	296	16	∣∣∣|1	∣∣∣|1	NOUN
ma-117	296	17	+	+	CCONJ
ma-117	296	18	θ1∆t|2	θ1∆t|2	X
ma-117	296	19	+	+	CCONJ
ma-117	296	20	∣∣θ2	∣∣θ2	VERB
ma-117	296	21	√	√	PROPN
ma-117	296	22	∆t	∆t	PROPN
ma-117	296	23	∣∣2∣∣∣	∣∣2∣∣∣	PROPN
ma-117	296	24	<	<	X
ma-117	296	25	1	1	NUM
ma-117	296	26	and	and	CCONJ
ma-117	296	27	lim	lim	PROPN
ma-117	296	28	∆t→0	∆t→0	PROPN
ma-117	296	29	(	(	PUNCT
ma-117	296	30	lim	lim	PROPN
ma-117	296	31	k→∞	k→∞	PROPN
ma-117	296	32	e	e	PROPN
ma-117	296	33	(	(	PUNCT
ma-117	296	34	|xk+1|2	|xk+1|2	PROPN
ma-117	296	35	)	)	PUNCT
ma-117	296	36	)	)	PUNCT
ma-117	297	1	=	=	SYM
ma-117	297	2	0	0	NUM
ma-117	297	3	proof	proof	NOUN
ma-117	297	4	.	.	PUNCT
ma-117	298	1	the	the	DET
ma-117	298	2	mean	mean	ADJ
ma-117	298	3	-	-	PUNCT
ma-117	298	4	square	square	NOUN
ma-117	298	5	of	of	ADP
ma-117	298	6	the	the	DET
ma-117	298	7	expression	expression	NOUN
ma-117	298	8	(	(	PUNCT
ma-117	298	9	4.6	4.6	NUM
ma-117	298	10	)	)	PUNCT
ma-117	298	11	gave	give	VERB
ma-117	298	12	:	:	PUNCT
ma-117	298	13	e	e	X
ma-117	299	1	[	[	X
ma-117	299	2	∣∣xemk+1	∣∣xemk+1	VERB
ma-117	299	3	∣∣2	∣∣2	NOUN
ma-117	299	4	]	]	X
ma-117	299	5	=	=	SYM
ma-117	299	6	e	e	X
ma-117	299	7	[	[	X
ma-117	299	8	∣∣∣xk(1	∣∣∣xk(1	NOUN
ma-117	299	9	+	+	CCONJ
ma-117	299	10	θ1∆t	θ1∆t	NOUN
ma-117	299	11	)	)	PUNCT
ma-117	300	1	+	+	NUM
ma-117	300	2	θ2xk	θ2xk	PUNCT
ma-117	300	3	√	√	ADP
ma-117	300	4	∆tzk	∆tzk	NOUN
ma-117	300	5	∣∣∣2	∣∣∣2	NOUN
ma-117	300	6	]	]	X
ma-117	300	7	=	=	SYM
ma-117	300	8	e	e	X
ma-117	300	9	[	[	PUNCT
ma-117	300	10	|xk(1	|xk(1	NOUN
ma-117	300	11	+	+	NUM
ma-117	300	12	θ1∆t)|2	θ1∆t)|2	PROPN
ma-117	300	13	+	+	CCONJ
ma-117	300	14	∣∣∣θ2	∣∣∣θ2	PROPN
ma-117	300	15	√	√	ADP
ma-117	300	16	∆txkzk	∆txkzk	ADJ
ma-117	300	17	∣∣∣2	∣∣∣2	NOUN
ma-117	300	18	+	+	CCONJ
ma-117	300	19	2	2	NUM
ma-117	300	20	∣∣∣xk(1	∣∣∣xk(1	NOUN
ma-117	300	21	+	+	CCONJ
ma-117	300	22	θ1∆t)θ2	θ1∆t)θ2	NOUN
ma-117	300	23	√	√	ADV
ma-117	300	24	∆txkzk	∆txkzk	ADJ
ma-117	300	25	∣∣∣	∣∣∣	ADJ
ma-117	300	26	]	]	PUNCT
ma-117	300	27	=	=	SYM
ma-117	300	28	e	e	X
ma-117	300	29	[	[	PUNCT
ma-117	300	30	|xk(1	|xk(1	NOUN
ma-117	300	31	+	+	CCONJ
ma-117	300	32	θ1∆t)|2	θ1∆t)|2	NOUN
ma-117	300	33	]	]	PUNCT
ma-117	301	1	+	+	PUNCT
ma-117	301	2	e	e	X
ma-117	302	1	[	[	X
ma-117	302	2	∣∣∣θ2	∣∣∣θ2	PROPN
ma-117	302	3	√	√	ADP
ma-117	302	4	∆txkzk	∆txkzk	ADJ
ma-117	302	5	∣∣∣2]+	∣∣∣2]+	NOUN
ma-117	302	6	2e	2e	NOUN
ma-117	302	7	[	[	PUNCT
ma-117	302	8	|xk(1	|xk(1	NOUN
ma-117	302	9	+	+	CCONJ
ma-117	302	10	θ1∆t)|	θ1∆t)|	ADJ
ma-117	302	11	∣∣∣θ2	∣∣∣θ2	PROPN
ma-117	302	12	√	√	ADP
ma-117	302	13	∆txkzk	∆txkzk	ADJ
ma-117	302	14	∣∣∣	∣∣∣	ADJ
ma-117	302	15	]	]	X
ma-117	302	16	=	=	PUNCT
ma-117	302	17	|(1	|(1	PROPN
ma-117	302	18	+	+	NUM
ma-117	302	19	θ1∆t)|2	θ1∆t)|2	PROPN
ma-117	302	20	e	e	NOUN
ma-117	302	21	[	[	PUNCT
ma-117	302	22	|xk	|xk	X
ma-117	302	23	|2	|2	NUM
ma-117	302	24	]	]	PUNCT
ma-117	302	25	+	+	CCONJ
ma-117	302	26	∣∣∣θ2	∣∣∣θ2	PROPN
ma-117	302	27	√	√	NUM
ma-117	302	28	∆t	∆t	PROPN
ma-117	302	29	∣∣∣2	∣∣∣2	NOUN
ma-117	302	30	e	e	PROPN
ma-117	303	1	[	[	X
ma-117	303	2	|xk	|xk	X
ma-117	303	3	|2	|2	NUM
ma-117	303	4	]	]	X
ma-117	303	5	=	=	SYM
ma-117	303	6	(	(	PUNCT
ma-117	303	7	|(1	|(1	NOUN
ma-117	303	8	+	+	CCONJ
ma-117	303	9	θ1∆t)|2	θ1∆t)|2	NOUN
ma-117	303	10	+	+	CCONJ
ma-117	303	11	∣∣∣θ2	∣∣∣θ2	PROPN
ma-117	303	12	√	√	CCONJ
ma-117	303	13	∆t	∆t	NOUN
ma-117	303	14	∣∣∣2)e	∣∣∣2)e	NOUN
ma-117	303	15	[	[	X
ma-117	303	16	|xt	|xt	NUM
ma-117	303	17	|2	|2	NUM
ma-117	303	18	]	]	X
ma-117	303	19	=	=	SYM
ma-117	303	20	(	(	PUNCT
ma-117	303	21	|(1	|(1	NOUN
ma-117	303	22	+	+	CCONJ
ma-117	303	23	θ1∆t)|2	θ1∆t)|2	NOUN
ma-117	303	24	+	+	CCONJ
ma-117	303	25	∣∣∣θ2	∣∣∣θ2	PROPN
ma-117	303	26	√	√	ADP
ma-117	303	27	∆t	∆t	PROPN
ma-117	303	28	∣∣∣2)(|(1	∣∣∣2)(|(1	PROPN
ma-117	303	29	+	+	CCONJ
ma-117	303	30	θ1∆t)|2	θ1∆t)|2	PROPN
ma-117	303	31	+	+	CCONJ
ma-117	303	32	∣∣∣θ2	∣∣∣θ2	PROPN
ma-117	303	33	√	√	CCONJ
ma-117	303	34	∆t	∆t	NOUN
ma-117	303	35	∣∣∣2)e	∣∣∣2)e	NOUN
ma-117	303	36	[	[	X
ma-117	303	37	|xk−1|2	|xk−1|2	X
ma-117	303	38	]	]	PUNCT
ma-117	303	39	...	...	PUNCT
ma-117	304	1	=	=	PUNCT
ma-117	304	2	(	(	PUNCT
ma-117	304	3	|(1	|(1	NOUN
ma-117	304	4	+	+	CCONJ
ma-117	304	5	θ1∆t)|2	θ1∆t)|2	NOUN
ma-117	304	6	+	+	CCONJ
ma-117	304	7	∣∣∣θ2	∣∣∣θ2	PROPN
ma-117	304	8	√	√	ADP
ma-117	304	9	∆t	∆t	PROPN
ma-117	304	10	∣∣∣2)2k+2	∣∣∣2)2k+2	NUM
ma-117	304	11	e	e	NOUN
ma-117	304	12	[	[	PUNCT
ma-117	304	13	|x0|2	|x0|2	PROPN
ma-117	304	14	]	]	PUNCT
ma-117	304	15	we	we	PRON
ma-117	304	16	get	get	VERB
ma-117	304	17	a	a	DET
ma-117	304	18	geometric	geometric	ADJ
ma-117	304	19	sequence	sequence	NOUN
ma-117	304	20	:	:	PUNCT
ma-117	304	21	e	e	X
ma-117	304	22	(	(	PUNCT
ma-117	304	23	∣∣xemk+1	∣∣xemk+1	VERB
ma-117	304	24	∣∣2	∣∣2	NUM
ma-117	304	25	)	)	PUNCT
ma-117	304	26	=	=	NOUN
ma-117	305	1	(	(	PUNCT
ma-117	305	2	|(1	|(1	NOUN
ma-117	305	3	+	+	CCONJ
ma-117	305	4	θ1∆t)|2	θ1∆t)|2	NOUN
ma-117	305	5	+	+	CCONJ
ma-117	305	6	∣∣∣θ2	∣∣∣θ2	PROPN
ma-117	305	7	√	√	ADJ
ma-117	305	8	∆t	∆t	NOUN
ma-117	305	9	∣∣∣2)2(k+1	∣∣∣2)2(k+1	PROPN
ma-117	305	10	)	)	PUNCT
ma-117	305	11	e	e	NOUN
ma-117	305	12	(	(	PUNCT
ma-117	305	13	|x0|2	|x0|2	PROPN
ma-117	305	14	)	)	PUNCT
ma-117	305	15	for	for	ADP
ma-117	305	16	∣∣∣|1	∣∣∣|1	NOUN
ma-117	305	17	+	+	CCONJ
ma-117	305	18	θ1∆t|2	θ1∆t|2	X
ma-117	305	19	+	+	CCONJ
ma-117	305	20	∣∣θ2	∣∣θ2	VERB
ma-117	305	21	√	√	PROPN
ma-117	305	22	∆t	∆t	PROPN
ma-117	305	23	∣∣2∣∣∣	∣∣2∣∣∣	PROPN
ma-117	305	24	<	<	X
ma-117	305	25	1	1	NUM
ma-117	305	26	the	the	DET
ma-117	305	27	sequence	sequence	NOUN
ma-117	305	28	converges	converge	VERB
ma-117	305	29	,	,	PUNCT
ma-117	305	30	and	and	CCONJ
ma-117	305	31	passing	pass	VERB
ma-117	305	32	to	to	ADP
ma-117	305	33	the	the	DET
ma-117	305	34	limit	limit	NOUN
ma-117	305	35	,	,	PUNCT
ma-117	305	36	we	we	PRON
ma-117	305	37	obtain	obtain	VERB
ma-117	305	38	fora	forum	NOUN
ma-117	305	39	∀∆t	∀∆t	X
ma-117	305	40	→	→	SYM
ma-117	305	41	0	0	NUM
ma-117	305	42	and	and	CCONJ
ma-117	305	43	k	k	PROPN
ma-117	305	44	→	→	SYM
ma-117	305	45	+	+	PROPN
ma-117	305	46	∞	∞	PROPN
ma-117	305	47	,	,	PUNCT
ma-117	305	48	lim	lim	PROPN
ma-117	305	49	∆t→0	∆t→0	PROPN
ma-117	305	50	(	(	PUNCT
ma-117	305	51	lim	lim	PROPN
ma-117	305	52	k→∞	k→∞	PROPN
ma-117	305	53	e	e	PROPN
ma-117	305	54	(	(	PUNCT
ma-117	305	55	|xk+1|2	|xk+1|2	PROPN
ma-117	305	56	)	)	PUNCT
ma-117	305	57	)	)	PUNCT
ma-117	306	1	=	=	SYM
ma-117	306	2	0	0	NUM
ma-117	306	3	�	�	PROPN
ma-117	306	4	https://doi.org/10.28924/ada/ma.3.8	https://doi.org/10.28924/ada/ma.3.8	PROPN
ma-117	306	5	eur	eur	PROPN
ma-117	306	6	.	.	PUNCT
ma-117	307	1	j.	j.	PROPN
ma-117	307	2	math	math	PROPN
ma-117	307	3	.	.	PUNCT
ma-117	308	1	anal	anal	PROPN
ma-117	308	2	.	.	PUNCT
ma-117	309	1	10.28924	10.28924	NUM
ma-117	309	2	/	/	SYM
ma-117	309	3	ada	ada	PROPN
ma-117	309	4	/	/	SYM
ma-117	309	5	ma.3.8	ma.3.8	PROPN
ma-117	309	6	144.3	144.3	NUM
ma-117	309	7	.	.	PUNCT
ma-117	310	1	milshtein	milshtein	PROPN
ma-117	310	2	’s	’s	PART
ma-117	310	3	scheme	scheme	NOUN
ma-117	310	4	stabilities	stability	NOUN
ma-117	310	5	.	.	PUNCT
ma-117	311	1	the	the	DET
ma-117	311	2	milshtein	milshtein	PROPN
ma-117	311	3	schema	schema	NOUN
ma-117	311	4	associated	associate	VERB
ma-117	311	5	to	to	ADP
ma-117	311	6	the	the	DET
ma-117	311	7	expression(4.1	expression(4.1	ADJ
ma-117	311	8	)	)	PUNCT
ma-117	311	9	isgiven	isgiven	VERB
ma-117	311	10	by	by	ADP
ma-117	311	11	:	:	PUNCT
ma-117	311	12	xmk+1	xmk+1	PROPN
ma-117	311	13	=	=	PUNCT
ma-117	311	14	xk	xk	PROPN
ma-117	312	1	+	+	CCONJ
ma-117	312	2	b(xk)∆t	b(xk)∆t	PROPN
ma-117	312	3	+	+	X
ma-117	312	4	σ(xk)∆bk	σ(xk)∆bk	NOUN
ma-117	312	5	+	+	CCONJ
ma-117	312	6	1	1	NUM
ma-117	312	7	2	2	NUM
ma-117	312	8	σσ′(xk	σσ′(xk	NOUN
ma-117	312	9	)	)	PUNCT
ma-117	312	10	{	{	PUNCT
ma-117	312	11	(	(	PUNCT
ma-117	312	12	∆bk)2	∆bk)2	PUNCT
ma-117	312	13	−	−	PROPN
ma-117	312	14	∆t	∆t	PROPN
ma-117	312	15	}	}	PUNCT
ma-117	312	16	=	=	SYM
ma-117	312	17	xk	xk	PROPN
ma-117	313	1	+	+	CCONJ
ma-117	313	2	θ1xk∆t	θ1xk∆t	NOUN
ma-117	313	3	+	+	CCONJ
ma-117	313	4	θ2xk∆bk	θ2xk∆bk	ADV
ma-117	313	5	+	+	CCONJ
ma-117	313	6	1	1	NUM
ma-117	313	7	2	2	NUM
ma-117	313	8	θ2xtθ2	θ2xtθ2	NOUN
ma-117	313	9	{	{	PUNCT
ma-117	313	10	(	(	PUNCT
ma-117	313	11	∆bk)2	∆bk)2	PUNCT
ma-117	313	12	−	−	PROPN
ma-117	313	13	∆t	∆t	PROPN
ma-117	313	14	}	}	PUNCT
ma-117	313	15	=	=	SYM
ma-117	313	16	xk	xk	PROPN
ma-117	314	1	+	+	CCONJ
ma-117	314	2	θ1xk∆t	θ1xk∆t	VERB
ma-117	314	3	+	+	NUM
ma-117	314	4	θ2xk	θ2xk	NOUN
ma-117	314	5	√	√	ADP
ma-117	314	6	∆tzk	∆tzk	NOUN
ma-117	314	7	+	+	CCONJ
ma-117	314	8	1	1	NUM
ma-117	314	9	2	2	NUM
ma-117	314	10	θ2	θ2	ADP
ma-117	314	11	2xk	2xk	NOUN
ma-117	314	12	(	(	PUNCT
ma-117	314	13	∆tz2	∆tz2	PROPN
ma-117	314	14	k	k	PROPN
ma-117	314	15	−	−	PROPN
ma-117	314	16	∆t	∆t	PROPN
ma-117	314	17	)	)	PUNCT
ma-117	315	1	=	=	PUNCT
ma-117	315	2	(	(	PUNCT
ma-117	315	3	1	1	NUM
ma-117	315	4	+	+	NUM
ma-117	315	5	θ1∆t	θ1∆t	NOUN
ma-117	315	6	−	−	NUM
ma-117	315	7	1	1	NUM
ma-117	315	8	2	2	NUM
ma-117	315	9	θ2	θ2	ADP
ma-117	315	10	2∆t	2∆t	NUM
ma-117	315	11	)	)	PUNCT
ma-117	315	12	xk	xk	PROPN
ma-117	316	1	+	+	PUNCT
ma-117	316	2	θ2xk	θ2xk	PUNCT
ma-117	316	3	√	√	ADP
ma-117	316	4	∆tzk	∆tzk	NOUN
ma-117	316	5	+	+	CCONJ
ma-117	316	6	1	1	NUM
ma-117	316	7	2	2	NUM
ma-117	316	8	θ2	θ2	PROPN
ma-117	316	9	2xk∆tz2	2xk∆tz2	NOUN
ma-117	316	10	k	k	X
ma-117	316	11	=	=	PUNCT
ma-117	316	12	(	(	PUNCT
ma-117	316	13	1	1	NUM
ma-117	316	14	+	+	CCONJ
ma-117	316	15	(	(	PUNCT
ma-117	316	16	θ1	θ1	NOUN
ma-117	316	17	−	−	PROPN
ma-117	316	18	1	1	NUM
ma-117	316	19	2	2	NUM
ma-117	316	20	θ2	θ2	ADP
ma-117	316	21	2	2	NUM
ma-117	316	22	)	)	PUNCT
ma-117	316	23	∆t	∆t	PROPN
ma-117	316	24	)	)	PUNCT
ma-117	316	25	xk	xk	PROPN
ma-117	317	1	+	+	PUNCT
ma-117	317	2	θ2xk	θ2xk	PUNCT
ma-117	317	3	√	√	ADP
ma-117	317	4	∆tzk	∆tzk	NOUN
ma-117	317	5	+	+	CCONJ
ma-117	317	6	1	1	NUM
ma-117	317	7	2	2	NUM
ma-117	317	8	θ2	θ2	PROPN
ma-117	317	9	2xk∆tz2	2xk∆tz2	PROPN
ma-117	317	10	kwe	kwe	PROPN
ma-117	317	11	have	have	VERB
ma-117	317	12	after	after	ADP
ma-117	317	13	calculation	calculation	NOUN
ma-117	317	14	:	:	PUNCT
ma-117	317	15	xmk+1	xmk+1	NOUN
ma-117	317	16	=	=	SYM
ma-117	317	17	xk	xk	PROPN
ma-117	317	18	(	(	PUNCT
ma-117	317	19	1	1	NUM
ma-117	317	20	+	+	CCONJ
ma-117	317	21	(	(	PUNCT
ma-117	317	22	θ1	θ1	NOUN
ma-117	317	23	−	−	PROPN
ma-117	317	24	1	1	NUM
ma-117	317	25	2	2	NUM
ma-117	317	26	θ2	θ2	ADP
ma-117	317	27	2	2	NUM
ma-117	317	28	)	)	PUNCT
ma-117	317	29	∆t	∆t	PROPN
ma-117	317	30	)	)	PUNCT
ma-117	318	1	+	+	NUM
ma-117	318	2	θ2xk	θ2xk	NOUN
ma-117	318	3	√	√	ADP
ma-117	318	4	∆tzk	∆tzk	NOUN
ma-117	318	5	+	+	CCONJ
ma-117	318	6	1	1	NUM
ma-117	318	7	2	2	NUM
ma-117	318	8	θ2	θ2	PROPN
ma-117	318	9	2xk∆tz2	2xk∆tz2	PROPN
ma-117	318	10	k	k	PROPN
ma-117	318	11	(	(	PUNCT
ma-117	318	12	4.7	4.7	NUM
ma-117	318	13	)	)	PUNCT
ma-117	318	14	we	we	PRON
ma-117	318	15	now	now	ADV
ma-117	318	16	consider	consider	VERB
ma-117	318	17	the	the	DET
ma-117	318	18	same	same	ADJ
ma-117	318	19	model	model	NOUN
ma-117	318	20	of	of	ADP
ma-117	318	21	geometric	geometric	ADJ
ma-117	318	22	brownian	brownian	ADJ
ma-117	318	23	motion	motion	NOUN
ma-117	318	24	,	,	PUNCT
ma-117	318	25	we	we	PRON
ma-117	318	26	state	state	VERB
ma-117	318	27	some	some	DET
ma-117	318	28	results	result	NOUN
ma-117	318	29	on	on	ADP
ma-117	318	30	thestabilities	thestabilitie	NOUN
ma-117	318	31	following	follow	VERB
ma-117	318	32	milshtein	milshtein	PROPN
ma-117	318	33	’s	’s	PART
ma-117	318	34	scheme	scheme	NOUN
ma-117	318	35	and	and	CCONJ
ma-117	318	36	we	we	PRON
ma-117	318	37	prove	prove	VERB
ma-117	318	38	these	these	DET
ma-117	318	39	results	result	NOUN
ma-117	318	40	.	.	PUNCT
ma-117	319	1	4.3.1	4.3.1	X
ma-117	319	2	.	.	PUNCT
ma-117	320	1	mean	mean	VERB
ma-117	320	2	stability	stability	NOUN
ma-117	320	3	of	of	ADP
ma-117	320	4	milshtein	milshtein	PROPN
ma-117	320	5	’s	’s	PART
ma-117	320	6	scheme	scheme	NOUN
ma-117	320	7	.	.	PUNCT
ma-117	321	1	theorem	theorem	VERB
ma-117	321	2	4.3	4.3	NUM
ma-117	321	3	.	.	PUNCT
ma-117	322	1	(	(	PUNCT
ma-117	322	2	mean	mean	VERB
ma-117	322	3	stability	stability	NOUN
ma-117	322	4	of	of	ADP
ma-117	322	5	milshtein	milshtein	PROPN
ma-117	322	6	’s	’s	PART
ma-117	322	7	scheme	scheme	NOUN
ma-117	322	8	)	)	PUNCT
ma-117	322	9	the	the	DET
ma-117	322	10	milshtein	milshtein	PROPN
ma-117	322	11	’s	’s	PART
ma-117	322	12	scheme	scheme	NOUN
ma-117	322	13	(	(	PUNCT
ma-117	322	14	4.7	4.7	NUM
ma-117	322	15	)	)	PUNCT
ma-117	322	16	assocated	assocate	VERB
ma-117	322	17	to(4.1	to(4.1	NOUN
ma-117	322	18	)	)	PUNCT
ma-117	322	19	model	model	NOUN
ma-117	322	20	is	be	AUX
ma-117	322	21	mean	mean	ADV
ma-117	322	22	asymptotically	asymptotically	ADV
ma-117	322	23	stable	stable	ADJ
ma-117	322	24	if	if	SCONJ
ma-117	322	25	e	e	PROPN
ma-117	322	26	(	(	PUNCT
ma-117	322	27	xmk+1	xmk+1	X
ma-117	322	28	)	)	PUNCT
ma-117	323	1	=	=	PUNCT
ma-117	324	1	[	[	X
ma-117	324	2	1	1	NUM
ma-117	324	3	+	+	NUM
ma-117	324	4	θ1∆t]k+1	θ1∆t]k+1	NOUN
ma-117	324	5	e	e	X
ma-117	324	6	(	(	PUNCT
ma-117	324	7	x0	x0	PROPN
ma-117	324	8	)	)	PUNCT
ma-117	324	9	with	with	ADP
ma-117	324	10	|1	|1	PRON
ma-117	324	11	+	+	NUM
ma-117	324	12	θ1∆t|	θ1∆t|	PROPN
ma-117	324	13	<	<	X
ma-117	324	14	1	1	NUM
ma-117	324	15	and	and	CCONJ
ma-117	324	16	lim	lim	PROPN
ma-117	324	17	∆t→0	∆t→0	PROPN
ma-117	324	18	(	(	PUNCT
ma-117	324	19	lim	lim	PROPN
ma-117	324	20	k→∞	k→∞	PROPN
ma-117	324	21	e	e	PROPN
ma-117	324	22	(	(	PUNCT
ma-117	324	23	xmk+1	xmk+1	X
ma-117	324	24	)	)	PUNCT
ma-117	324	25	)	)	PUNCT
ma-117	325	1	=	=	SYM
ma-117	325	2	0	0	NUM
ma-117	326	1	proof	proof	NOUN
ma-117	326	2	.	.	PUNCT
ma-117	327	1	applying	apply	VERB
ma-117	327	2	the	the	DET
ma-117	327	3	usual	usual	ADJ
ma-117	327	4	approach	approach	NOUN
ma-117	327	5	,	,	PUNCT
ma-117	327	6	let	let	VERB
ma-117	327	7	us	we	PRON
ma-117	327	8	evaluate	evaluate	VERB
ma-117	327	9	the	the	DET
ma-117	327	10	mean	mean	NOUN
ma-117	327	11	of	of	ADP
ma-117	327	12	gives	give	NOUN
ma-117	327	13	:	:	PUNCT
ma-117	327	14	e	e	X
ma-117	327	15	(	(	PUNCT
ma-117	327	16	xmk+1	xmk+1	X
ma-117	327	17	)	)	PUNCT
ma-117	328	1	=	=	PUNCT
ma-117	328	2	e	e	X
ma-117	328	3	(	(	PUNCT
ma-117	328	4	xk	xk	X
ma-117	328	5	(	(	PUNCT
ma-117	328	6	1	1	NUM
ma-117	328	7	+	+	CCONJ
ma-117	328	8	(	(	PUNCT
ma-117	328	9	θ1	θ1	NOUN
ma-117	328	10	−	−	PROPN
ma-117	328	11	1	1	NUM
ma-117	328	12	2	2	NUM
ma-117	328	13	θ2	θ2	ADP
ma-117	328	14	2	2	NUM
ma-117	328	15	)	)	PUNCT
ma-117	328	16	∆t	∆t	PROPN
ma-117	328	17	)	)	PUNCT
ma-117	329	1	+	+	NUM
ma-117	329	2	θ2xk	θ2xk	NOUN
ma-117	329	3	√	√	ADP
ma-117	329	4	∆tzk	∆tzk	NOUN
ma-117	329	5	+	+	CCONJ
ma-117	329	6	1	1	NUM
ma-117	329	7	2	2	NUM
ma-117	329	8	θ2	θ2	PROPN
ma-117	329	9	2xk∆tz2	2xk∆tz2	PROPN
ma-117	329	10	k	k	PROPN
ma-117	329	11	)	)	PUNCT
ma-117	330	1	=	=	SYM
ma-117	330	2	e	e	X
ma-117	330	3	(	(	PUNCT
ma-117	330	4	xk	xk	X
ma-117	330	5	(	(	PUNCT
ma-117	330	6	1	1	NUM
ma-117	330	7	+	+	CCONJ
ma-117	330	8	(	(	PUNCT
ma-117	330	9	θ1	θ1	NOUN
ma-117	330	10	−	−	PROPN
ma-117	330	11	1	1	NUM
ma-117	330	12	2	2	NUM
ma-117	330	13	θ2	θ2	ADP
ma-117	330	14	2	2	NUM
ma-117	330	15	)	)	PUNCT
ma-117	330	16	∆t	∆t	PROPN
ma-117	330	17	)	)	PUNCT
ma-117	330	18	)	)	PUNCT
ma-117	331	1	+	+	CCONJ
ma-117	331	2	e	e	X
ma-117	331	3	(	(	PUNCT
ma-117	331	4	θ2xk	θ2xk	PUNCT
ma-117	331	5	√	√	ADP
ma-117	331	6	∆tzk	∆tzk	NOUN
ma-117	331	7	)	)	PUNCT
ma-117	332	1	+	+	CCONJ
ma-117	332	2	e	e	X
ma-117	332	3	(	(	PUNCT
ma-117	332	4	1	1	NUM
ma-117	332	5	2	2	NUM
ma-117	332	6	θ2	θ2	PROPN
ma-117	332	7	2xk∆tz2	2xk∆tz2	PROPN
ma-117	332	8	k	k	NOUN
ma-117	332	9	)	)	PUNCT
ma-117	333	1	=	=	PUNCT
ma-117	333	2	(	(	PUNCT
ma-117	333	3	1	1	NUM
ma-117	333	4	+	+	CCONJ
ma-117	333	5	(	(	PUNCT
ma-117	333	6	θ1	θ1	NOUN
ma-117	333	7	−	−	PROPN
ma-117	333	8	1	1	NUM
ma-117	333	9	2	2	NUM
ma-117	333	10	θ2	θ2	ADP
ma-117	333	11	2	2	NUM
ma-117	333	12	)	)	PUNCT
ma-117	333	13	∆t	∆t	PROPN
ma-117	334	1	+	+	CCONJ
ma-117	334	2	1	1	NUM
ma-117	334	3	2	2	NUM
ma-117	334	4	θ2	θ2	ADP
ma-117	334	5	2∆t	2∆t	NUM
ma-117	334	6	)	)	PUNCT
ma-117	335	1	e	e	X
ma-117	335	2	(	(	PUNCT
ma-117	335	3	xk	xk	PROPN
ma-117	335	4	)	)	PUNCT
ma-117	335	5	=	=	SYM
ma-117	335	6	(	(	PUNCT
ma-117	335	7	1	1	NUM
ma-117	335	8	+	+	NUM
ma-117	335	9	θ1∆t)e	θ1∆t)e	NUM
ma-117	335	10	(	(	PUNCT
ma-117	335	11	xk	xk	NOUN
ma-117	335	12	)	)	PUNCT
ma-117	335	13	...	...	PUNCT
ma-117	336	1	=	=	PUNCT
ma-117	336	2	(	(	PUNCT
ma-117	336	3	1	1	NUM
ma-117	336	4	+	+	NUM
ma-117	336	5	θ1∆t)k+1	θ1∆t)k+1	ADJ
ma-117	336	6	e	e	X
ma-117	336	7	(	(	PUNCT
ma-117	336	8	x0	x0	PROPN
ma-117	336	9	)	)	PUNCT
ma-117	336	10	https://doi.org/10.28924/ada/ma.3.8	https://doi.org/10.28924/ada/ma.3.8	NUM
ma-117	336	11	eur	eur	NOUN
ma-117	336	12	.	.	PUNCT
ma-117	337	1	j.	j.	PROPN
ma-117	337	2	math	math	PROPN
ma-117	337	3	.	.	PUNCT
ma-117	338	1	anal	anal	PROPN
ma-117	338	2	.	.	PUNCT
ma-117	339	1	10.28924	10.28924	NUM
ma-117	339	2	/	/	SYM
ma-117	339	3	ada	ada	PROPN
ma-117	339	4	/	/	SYM
ma-117	339	5	ma.3.8	ma.3.8	PROPN
ma-117	339	6	15ultimately	15ultimately	ADV
ma-117	339	7	we	we	PRON
ma-117	339	8	get	get	VERB
ma-117	339	9	that	that	PRON
ma-117	339	10	:	:	PUNCT
ma-117	340	1	e	e	X
ma-117	340	2	(	(	PUNCT
ma-117	340	3	xmk+1	xmk+1	X
ma-117	340	4	)	)	PUNCT
ma-117	340	5	=	=	PUNCT
ma-117	341	1	[	[	X
ma-117	341	2	1	1	NUM
ma-117	341	3	+	+	NUM
ma-117	341	4	θ1∆t]k+1	θ1∆t]k+1	NOUN
ma-117	341	5	e	e	NOUN
ma-117	341	6	(	(	PUNCT
ma-117	341	7	x0)as	x0)as	PROPN
ma-117	341	8	the	the	DET
ma-117	341	9	previous	previous	ADJ
ma-117	341	10	expression	expression	NOUN
ma-117	341	11	has	have	AUX
ma-117	341	12	the	the	DET
ma-117	341	13	form	form	NOUN
ma-117	341	14	of	of	ADP
ma-117	341	15	a	a	DET
ma-117	341	16	geometric	geometric	ADJ
ma-117	341	17	sequence	sequence	NOUN
ma-117	341	18	,	,	PUNCT
ma-117	341	19	we	we	PRON
ma-117	341	20	know	know	VERB
ma-117	341	21	that	that	SCONJ
ma-117	341	22	it	it	PRON
ma-117	341	23	converges	converge	VERB
ma-117	341	24	if	if	SCONJ
ma-117	341	25	|1	|1	PRON
ma-117	342	1	+	+	NUM
ma-117	342	2	θ1∆t|	θ1∆t|	PROPN
ma-117	342	3	<	<	X
ma-117	342	4	1	1	NUM
ma-117	342	5	,	,	PUNCT
ma-117	342	6	passing	pass	VERB
ma-117	342	7	to	to	ADP
ma-117	342	8	the	the	DET
ma-117	342	9	limit	limit	NOUN
ma-117	342	10	,	,	PUNCT
ma-117	342	11	for	for	ADP
ma-117	342	12	all	all	DET
ma-117	342	13	∆t	∆t	PROPN
ma-117	342	14	→	→	SYM
ma-117	342	15	0	0	NUM
ma-117	342	16	and	and	CCONJ
ma-117	342	17	k	k	PROPN
ma-117	342	18	→	→	PUNCT
ma-117	342	19	+	+	NOUN
ma-117	342	20	∞	∞	NOUN
ma-117	342	21	we	we	PRON
ma-117	342	22	find	find	VERB
ma-117	342	23	the	the	DET
ma-117	342	24	results	result	NOUN
ma-117	342	25	searched	search	VERB
ma-117	342	26	i.e	i.e	PRON
ma-117	342	27	:	:	PUNCT
ma-117	342	28	lim	lim	PROPN
ma-117	342	29	∆t→0	∆t→0	PROPN
ma-117	342	30	(	(	PUNCT
ma-117	342	31	lim	lim	PROPN
ma-117	342	32	k→∞	k→∞	PROPN
ma-117	342	33	e	e	PROPN
ma-117	342	34	(	(	PUNCT
ma-117	342	35	xmk+1	xmk+1	X
ma-117	342	36	)	)	PUNCT
ma-117	342	37	)	)	PUNCT
ma-117	343	1	=	=	SYM
ma-117	343	2	0	0	NUM
ma-117	343	3	�	�	PROPN
ma-117	343	4	4.3.2	4.3.2	NOUN
ma-117	343	5	.	.	PUNCT
ma-117	344	1	mean	mean	ADJ
ma-117	344	2	-	-	PUNCT
ma-117	344	3	square	square	ADJ
ma-117	344	4	stability	stability	NOUN
ma-117	344	5	of	of	ADP
ma-117	344	6	milshtein	milshtein	PROPN
ma-117	344	7	’s	’s	PART
ma-117	344	8	scheme	scheme	NOUN
ma-117	344	9	.	.	PUNCT
ma-117	345	1	theorem	theorem	VERB
ma-117	345	2	4.4	4.4	NUM
ma-117	345	3	.	.	PUNCT
ma-117	346	1	(	(	PUNCT
ma-117	346	2	mean	mean	ADJ
ma-117	346	3	-	-	PUNCT
ma-117	346	4	square	square	ADJ
ma-117	346	5	stability	stability	NOUN
ma-117	346	6	of	of	ADP
ma-117	346	7	milshtein	milshtein	PROPN
ma-117	346	8	’s	’s	PART
ma-117	346	9	scheme	scheme	NOUN
ma-117	346	10	)	)	PUNCT
ma-117	346	11	the	the	DET
ma-117	346	12	milshtein	milshtein	PROPN
ma-117	346	13	scheme	scheme	NOUN
ma-117	346	14	(	(	PUNCT
ma-117	346	15	4.7	4.7	NUM
ma-117	346	16	)	)	PUNCT
ma-117	346	17	associated	associate	VERB
ma-117	346	18	to	to	ADP
ma-117	346	19	(	(	PUNCT
ma-117	346	20	4.1	4.1	NUM
ma-117	346	21	)	)	PUNCT
ma-117	346	22	model	model	NOUN
ma-117	346	23	is	be	AUX
ma-117	346	24	mean	mean	ADJ
ma-117	346	25	-	-	PUNCT
ma-117	346	26	square	square	ADJ
ma-117	346	27	asymptotically	asymptotically	ADV
ma-117	346	28	stable	stable	ADJ
ma-117	346	29	if	if	SCONJ
ma-117	346	30	e	e	PROPN
ma-117	346	31	(	(	PUNCT
ma-117	346	32	∣∣xmk+1	∣∣xmk+1	X
ma-117	346	33	∣∣	∣∣	X
ma-117	346	34	)	)	PUNCT
ma-117	346	35	=	=	PUNCT
ma-117	347	1	[	[	X
ma-117	347	2	∣∣∣∣1	∣∣∣∣1	NOUN
ma-117	347	3	+	+	CCONJ
ma-117	347	4	(	(	PUNCT
ma-117	347	5	θ1	θ1	NOUN
ma-117	347	6	−	−	PROPN
ma-117	347	7	1	1	NUM
ma-117	347	8	2	2	NUM
ma-117	347	9	θ2	θ2	ADP
ma-117	347	10	2	2	NUM
ma-117	347	11	)	)	PUNCT
ma-117	347	12	∆t	∆t	VERB
ma-117	347	13	∣∣∣∣2	∣∣∣∣2	NOUN
ma-117	347	14	+	+	CCONJ
ma-117	347	15	∣∣∣θ2	∣∣∣θ2	PROPN
ma-117	347	16	√	√	ADP
ma-117	347	17	∆t	∆t	PROPN
ma-117	347	18	∣∣∣2	∣∣∣2	NOUN
ma-117	347	19	+	+	CCONJ
ma-117	347	20	∣∣∣∣12θ2	∣∣∣∣12θ2	NUM
ma-117	347	21	2∆t	2∆t	NUM
ma-117	347	22	∣∣∣∣2	∣∣∣∣2	NOUN
ma-117	347	23	]	]	PUNCT
ma-117	347	24	2(k+1	2(k+1	NUM
ma-117	347	25	)	)	PUNCT
ma-117	347	26	e	e	NOUN
ma-117	347	27	(	(	PUNCT
ma-117	347	28	|x0|2	|x0|2	PROPN
ma-117	347	29	)	)	PUNCT
ma-117	347	30	with	with	ADP
ma-117	347	31	∣∣∣∣∣1	∣∣∣∣∣1	VERB
ma-117	347	32	+	+	CCONJ
ma-117	347	33	(	(	PUNCT
ma-117	347	34	θ1	θ1	NOUN
ma-117	347	35	−	−	PROPN
ma-117	347	36	1	1	NUM
ma-117	347	37	2θ	2θ	NUM
ma-117	347	38	2	2	NUM
ma-117	347	39	2	2	NUM
ma-117	347	40	)	)	PUNCT
ma-117	347	41	∆t	∆t	PROPN
ma-117	347	42	∣∣2	∣∣2	PROPN
ma-117	347	43	+	+	CCONJ
ma-117	347	44	∣∣θ2	∣∣θ2	ADJ
ma-117	347	45	√	√	PROPN
ma-117	347	46	∆t	∆t	PROPN
ma-117	347	47	∣∣2	∣∣2	PROPN
ma-117	347	48	+	+	CCONJ
ma-117	347	49	∣∣1	∣∣1	NUM
ma-117	347	50	2θ	2θ	NUM
ma-117	347	51	2	2	NUM
ma-117	347	52	2∆t	2∆t	NUM
ma-117	347	53	∣∣2∣∣∣	∣∣2∣∣∣	PROPN
ma-117	347	54	<	<	X
ma-117	347	55	1	1	NUM
ma-117	347	56	and	and	CCONJ
ma-117	347	57	lim	lim	PROPN
ma-117	347	58	∆t→0	∆t→0	PROPN
ma-117	347	59	(	(	PUNCT
ma-117	347	60	lim	lim	PROPN
ma-117	347	61	k→∞	k→∞	PROPN
ma-117	347	62	e	e	PROPN
ma-117	347	63	(	(	PUNCT
ma-117	347	64	∣∣xmk+1	∣∣xmk+1	X
ma-117	347	65	∣∣2	∣∣2	NUM
ma-117	347	66	)	)	PUNCT
ma-117	347	67	)	)	PUNCT
ma-117	348	1	=	=	SYM
ma-117	348	2	0	0	NUM
ma-117	348	3	proof	proof	NOUN
ma-117	348	4	.	.	PUNCT
ma-117	349	1	let	let	VERB
ma-117	349	2	’s	’s	PRON
ma-117	349	3	start	start	VERB
ma-117	349	4	by	by	ADP
ma-117	349	5	calculating	calculate	VERB
ma-117	349	6	the	the	DET
ma-117	349	7	mean	mean	ADJ
ma-117	349	8	-	-	PUNCT
ma-117	349	9	sqaure	sqaure	NOUN
ma-117	349	10	of	of	ADP
ma-117	349	11	the	the	DET
ma-117	349	12	model	model	NOUN
ma-117	349	13	expression	expression	NOUN
ma-117	349	14	,	,	PUNCT
ma-117	349	15	ie	ie	X
ma-117	349	16	:	:	PUNCT
ma-117	349	17	e	e	X
ma-117	349	18	(	(	PUNCT
ma-117	349	19	∣∣xmk+1	∣∣xmk+1	X
ma-117	349	20	∣∣2	∣∣2	NUM
ma-117	349	21	)	)	PUNCT
ma-117	349	22	=	=	SYM
ma-117	349	23	e	e	X
ma-117	349	24	(	(	PUNCT
ma-117	349	25	∣∣∣∣xk	∣∣∣∣xk	PROPN
ma-117	349	26	(	(	PUNCT
ma-117	349	27	1	1	NUM
ma-117	349	28	+	+	CCONJ
ma-117	349	29	(	(	PUNCT
ma-117	349	30	θ1	θ1	NOUN
ma-117	349	31	−	−	PROPN
ma-117	349	32	1	1	NUM
ma-117	349	33	2	2	NUM
ma-117	349	34	θ2	θ2	ADP
ma-117	349	35	2	2	NUM
ma-117	349	36	)	)	PUNCT
ma-117	349	37	∆t	∆t	PROPN
ma-117	349	38	)	)	PUNCT
ma-117	350	1	+	+	NUM
ma-117	350	2	θ2xk	θ2xk	NOUN
ma-117	350	3	√	√	ADP
ma-117	350	4	∆tzk	∆tzk	NOUN
ma-117	350	5	+	+	CCONJ
ma-117	350	6	1	1	NUM
ma-117	350	7	2	2	NUM
ma-117	350	8	θ2	θ2	PROPN
ma-117	350	9	2xk∆tz2	2xk∆tz2	NOUN
ma-117	350	10	k	k	PROPN
ma-117	350	11	∣∣∣∣2	∣∣∣∣2	NOUN
ma-117	350	12	)	)	PUNCT
ma-117	351	1	=	=	SYM
ma-117	351	2	e	e	X
ma-117	351	3	(	(	PUNCT
ma-117	351	4	∣∣∣∣xk	∣∣∣∣xk	PROPN
ma-117	351	5	(	(	PUNCT
ma-117	351	6	1	1	NUM
ma-117	351	7	+	+	CCONJ
ma-117	351	8	(	(	PUNCT
ma-117	351	9	θ1	θ1	NOUN
ma-117	351	10	−	−	PROPN
ma-117	351	11	1	1	NUM
ma-117	351	12	2	2	NUM
ma-117	351	13	θ2	θ2	ADP
ma-117	351	14	2	2	NUM
ma-117	351	15	)	)	PUNCT
ma-117	351	16	∆t	∆t	PROPN
ma-117	351	17	)	)	PUNCT
ma-117	351	18	∣∣∣∣2	∣∣∣∣2	NOUN
ma-117	351	19	)	)	PUNCT
ma-117	352	1	+	+	CCONJ
ma-117	352	2	e	e	X
ma-117	352	3	(	(	PUNCT
ma-117	352	4	∣∣∣θ2xk	∣∣∣θ2xk	ADV
ma-117	352	5	√	√	ADP
ma-117	352	6	∆tzk	∆tzk	NOUN
ma-117	352	7	∣∣∣2)+	∣∣∣2)+	NOUN
ma-117	352	8	e	e	NOUN
ma-117	352	9	(	(	PUNCT
ma-117	352	10	∣∣∣∣12θ2	∣∣∣∣12θ2	NUM
ma-117	352	11	2xk∆tz2	2xk∆tz2	ADJ
ma-117	352	12	k	k	PROPN
ma-117	352	13	∣∣∣∣2	∣∣∣∣2	NOUN
ma-117	352	14	)	)	PUNCT
ma-117	353	1	=	=	SYM
ma-117	353	2	∣∣∣∣1	∣∣∣∣1	NOUN
ma-117	353	3	+	+	CCONJ
ma-117	353	4	(	(	PUNCT
ma-117	353	5	θ1	θ1	NOUN
ma-117	353	6	−	−	PROPN
ma-117	353	7	1	1	NUM
ma-117	353	8	2	2	NUM
ma-117	353	9	θ2	θ2	ADP
ma-117	353	10	2	2	NUM
ma-117	353	11	)	)	PUNCT
ma-117	353	12	∆t	∆t	VERB
ma-117	353	13	∣∣∣∣2	∣∣∣∣2	NOUN
ma-117	353	14	e	e	NOUN
ma-117	353	15	(	(	PUNCT
ma-117	353	16	|xk	|xk	X
ma-117	353	17	|2)+	|2)+	VERB
ma-117	353	18	∣∣∣θ2	∣∣∣θ2	PROPN
ma-117	353	19	√	√	NUM
ma-117	353	20	∆t	∆t	PROPN
ma-117	353	21	∣∣∣2	∣∣∣2	NOUN
ma-117	353	22	e	e	X
ma-117	353	23	(	(	PUNCT
ma-117	353	24	|xk	|xk	X
ma-117	353	25	|2)+	|2)+	VERB
ma-117	353	26	∣∣∣∣12θ2	∣∣∣∣12θ2	PROPN
ma-117	353	27	2∆t	2∆t	NUM
ma-117	353	28	∣∣∣∣2	∣∣∣∣2	NOUN
ma-117	353	29	e	e	NOUN
ma-117	353	30	(	(	PUNCT
ma-117	353	31	|xk	|xk	X
ma-117	353	32	|2	|2	NUM
ma-117	353	33	)	)	PUNCT
ma-117	353	34	=	=	NOUN
ma-117	354	1	[	[	X
ma-117	354	2	∣∣∣∣1	∣∣∣∣1	NOUN
ma-117	354	3	+	+	CCONJ
ma-117	354	4	(	(	PUNCT
ma-117	354	5	θ1	θ1	NOUN
ma-117	354	6	−	−	PROPN
ma-117	354	7	1	1	NUM
ma-117	354	8	2	2	NUM
ma-117	354	9	θ2	θ2	ADP
ma-117	354	10	2	2	NUM
ma-117	354	11	)	)	PUNCT
ma-117	354	12	∆t	∆t	VERB
ma-117	354	13	∣∣∣∣2	∣∣∣∣2	NOUN
ma-117	354	14	+	+	CCONJ
ma-117	354	15	∣∣∣θ2	∣∣∣θ2	PROPN
ma-117	354	16	√	√	ADP
ma-117	354	17	∆t	∆t	PROPN
ma-117	354	18	∣∣∣2	∣∣∣2	NOUN
ma-117	354	19	+	+	CCONJ
ma-117	354	20	∣∣∣∣12θ2	∣∣∣∣12θ2	NUM
ma-117	354	21	2∆t	2∆t	NUM
ma-117	354	22	∣∣∣∣2	∣∣∣∣2	NOUN
ma-117	354	23	]	]	PUNCT
ma-117	354	24	e	e	X
ma-117	354	25	(	(	PUNCT
ma-117	354	26	|xk	|xk	X
ma-117	354	27	|2	|2	NUM
ma-117	354	28	)	)	PUNCT
ma-117	354	29	...	...	PUNCT
ma-117	355	1	=	=	PUNCT
ma-117	356	1	[	[	X
ma-117	356	2	∣∣∣∣1	∣∣∣∣1	NOUN
ma-117	356	3	+	+	CCONJ
ma-117	356	4	(	(	PUNCT
ma-117	356	5	θ1	θ1	NOUN
ma-117	356	6	−	−	PROPN
ma-117	356	7	1	1	NUM
ma-117	356	8	2	2	NUM
ma-117	356	9	θ2	θ2	ADP
ma-117	356	10	2	2	NUM
ma-117	356	11	)	)	PUNCT
ma-117	356	12	∆t	∆t	VERB
ma-117	356	13	∣∣∣∣2	∣∣∣∣2	NOUN
ma-117	356	14	+	+	CCONJ
ma-117	356	15	∣∣∣θ2	∣∣∣θ2	PROPN
ma-117	356	16	√	√	ADP
ma-117	356	17	∆t	∆t	PROPN
ma-117	356	18	∣∣∣2	∣∣∣2	NOUN
ma-117	356	19	+	+	CCONJ
ma-117	356	20	∣∣∣∣12θ2	∣∣∣∣12θ2	NUM
ma-117	356	21	2∆t	2∆t	NUM
ma-117	356	22	∣∣∣∣2	∣∣∣∣2	NOUN
ma-117	356	23	]	]	PUNCT
ma-117	356	24	2(k+1	2(k+1	NUM
ma-117	356	25	)	)	PUNCT
ma-117	356	26	e	e	NOUN
ma-117	356	27	(	(	PUNCT
ma-117	356	28	|x0|2	|x0|2	X
ma-117	356	29	)	)	PUNCT
ma-117	356	30	continuing	continue	VERB
ma-117	356	31	with	with	ADP
ma-117	356	32	the	the	DET
ma-117	356	33	iterations	iteration	NOUN
ma-117	356	34	,	,	PUNCT
ma-117	356	35	we	we	PRON
ma-117	356	36	get	get	VERB
ma-117	356	37	:	:	PUNCT
ma-117	356	38	e	e	X
ma-117	356	39	(	(	PUNCT
ma-117	356	40	∣∣xmk+1	∣∣xmk+1	X
ma-117	356	41	∣∣	∣∣	X
ma-117	356	42	)	)	PUNCT
ma-117	356	43	=	=	PUNCT
ma-117	357	1	[	[	X
ma-117	357	2	∣∣∣∣1	∣∣∣∣1	NOUN
ma-117	357	3	+	+	CCONJ
ma-117	357	4	(	(	PUNCT
ma-117	357	5	θ1	θ1	NOUN
ma-117	357	6	−	−	PROPN
ma-117	357	7	1	1	NUM
ma-117	357	8	2	2	NUM
ma-117	357	9	θ2	θ2	ADP
ma-117	357	10	2	2	NUM
ma-117	357	11	)	)	PUNCT
ma-117	357	12	∆t	∆t	VERB
ma-117	357	13	∣∣∣∣2	∣∣∣∣2	NOUN
ma-117	357	14	+	+	CCONJ
ma-117	357	15	∣∣∣θ2	∣∣∣θ2	PROPN
ma-117	357	16	√	√	ADP
ma-117	357	17	∆t	∆t	PROPN
ma-117	357	18	∣∣∣2	∣∣∣2	NOUN
ma-117	357	19	+	+	CCONJ
ma-117	357	20	∣∣∣∣12θ2	∣∣∣∣12θ2	NUM
ma-117	357	21	2∆t	2∆t	NUM
ma-117	357	22	∣∣∣∣2	∣∣∣∣2	NOUN
ma-117	357	23	]	]	PUNCT
ma-117	357	24	2(k+1	2(k+1	NUM
ma-117	357	25	)	)	PUNCT
ma-117	357	26	e	e	NOUN
ma-117	357	27	(	(	PUNCT
ma-117	357	28	|x0|2	|x0|2	X
ma-117	357	29	)	)	PUNCT
ma-117	357	30	passing	pass	VERB
ma-117	357	31	to	to	ADP
ma-117	357	32	the	the	DET
ma-117	357	33	limit	limit	NOUN
ma-117	357	34	with	with	ADP
ma-117	357	35	∆t	∆t	PROPN
ma-117	357	36	→	→	SYM
ma-117	357	37	0	0	NUM
ma-117	357	38	and	and	CCONJ
ma-117	357	39	k	k	PROPN
ma-117	357	40	→	→	SYM
ma-117	357	41	+	+	NOUN
ma-117	357	42	∞	∞	PROPN
ma-117	357	43	,	,	PUNCT
ma-117	357	44	we	we	PRON
ma-117	357	45	obtain	obtain	VERB
ma-117	357	46	the	the	DET
ma-117	357	47	stated	state	VERB
ma-117	357	48	results	result	NOUN
ma-117	357	49	,	,	PUNCT
ma-117	357	50	ie	ie	ADJ
ma-117	357	51	:	:	PUNCT
ma-117	357	52	lim	lim	PROPN
ma-117	357	53	∆t→0	∆t→0	PROPN
ma-117	357	54	(	(	PUNCT
ma-117	357	55	lim	lim	PROPN
ma-117	357	56	k→∞	k→∞	PROPN
ma-117	357	57	e	e	PROPN
ma-117	357	58	(	(	PUNCT
ma-117	357	59	∣∣xmk+1	∣∣xmk+1	X
ma-117	357	60	∣∣2	∣∣2	NUM
ma-117	357	61	)	)	PUNCT
ma-117	357	62	)	)	PUNCT
ma-117	358	1	=	=	SYM
ma-117	358	2	0	0	NUM
ma-117	358	3	�	�	PROPN
ma-117	358	4	https://doi.org/10.28924/ada/ma.3.8	https://doi.org/10.28924/ada/ma.3.8	PROPN
ma-117	358	5	eur	eur	PROPN
ma-117	358	6	.	.	PUNCT
ma-117	359	1	j.	j.	PROPN
ma-117	359	2	math	math	PROPN
ma-117	359	3	.	.	PUNCT
ma-117	360	1	anal	anal	PROPN
ma-117	360	2	.	.	PUNCT
ma-117	361	1	10.28924	10.28924	NUM
ma-117	361	2	/	/	SYM
ma-117	361	3	ada	ada	PROPN
ma-117	361	4	/	/	SYM
ma-117	361	5	ma.3.8	ma.3.8	PROPN
ma-117	361	6	164.4	164.4	NUM
ma-117	361	7	.	.	PUNCT
ma-117	362	1	implicit	implicit	ADJ
ma-117	362	2	euler	euler	VERB
ma-117	362	3	-	-	PUNCT
ma-117	362	4	maruyama	maruyama	NOUN
ma-117	362	5	scheme	scheme	NOUN
ma-117	362	6	stabilities	stability	NOUN
ma-117	362	7	.	.	PUNCT
ma-117	363	1	the	the	DET
ma-117	363	2	implicit	implicit	ADJ
ma-117	363	3	euler	euler	VERB
ma-117	363	4	-	-	PUNCT
ma-117	363	5	maruyama	maruyama	NOUN
ma-117	363	6	scheme	scheme	NOUN
ma-117	363	7	(	(	PUNCT
ma-117	363	8	iem)gives	iem)give	NOUN
ma-117	363	9	:	:	PUNCT
ma-117	363	10	xiemk+1	xiemk+1	NOUN
ma-117	364	1	=	=	PUNCT
ma-117	364	2	xk	xk	PROPN
ma-117	365	1	+	+	CCONJ
ma-117	365	2	b(xk+1)∆t	b(xk+1)∆t	X
ma-117	365	3	+	+	CCONJ
ma-117	365	4	δ(xt)∆bk	δ(xt)∆bk	NOUN
ma-117	365	5	xk+1	xk+1	NOUN
ma-117	365	6	=	=	SYM
ma-117	365	7	xk	xk	PROPN
ma-117	366	1	+	+	CCONJ
ma-117	366	2	θ1xk+1∆t	θ1xk+1∆t	PROPN
ma-117	366	3	+	+	CCONJ
ma-117	366	4	θ2xt∆bk	θ2xt∆bk	NOUN
ma-117	366	5	xk+1	xk+1	NUM
ma-117	366	6	−	−	PROPN
ma-117	366	7	θ1xk+1∆t	θ1xk+1∆t	PROPN
ma-117	366	8	=	=	SYM
ma-117	366	9	xk	xk	PROPN
ma-117	367	1	+	+	CCONJ
ma-117	367	2	θ2xk∆bk	θ2xk∆bk	ADV
ma-117	367	3	xk+1	xk+1	NUM
ma-117	367	4	(	(	PUNCT
ma-117	367	5	1−	1−	NUM
ma-117	367	6	θ1∆t	θ1∆t	NOUN
ma-117	367	7	)	)	PUNCT
ma-117	367	8	=	=	SYM
ma-117	367	9	xk	xk	PROPN
ma-117	368	1	+	+	CCONJ
ma-117	368	2	θ2xk∆bkwe	θ2xk∆bkwe	VERB
ma-117	368	3	obtain	obtain	VERB
ma-117	368	4	:	:	PUNCT
ma-117	368	5	xiemk+1	xiemk+1	NOUN
ma-117	368	6	=	=	SYM
ma-117	368	7	1	1	NUM
ma-117	368	8	1−	1−	NUM
ma-117	368	9	θ1∆t	θ1∆t	NOUN
ma-117	368	10	xk	xk	PROPN
ma-117	369	1	+	+	CCONJ
ma-117	369	2	θ2	θ2	PROPN
ma-117	369	3	√	√	ADV
ma-117	369	4	∆t	∆t	PROPN
ma-117	369	5	1−	1−	NUM
ma-117	369	6	θ1∆t	θ1∆t	NOUN
ma-117	369	7	xkzk	xkzk	PROPN
ma-117	369	8	zk	zk	PROPN
ma-117	369	9	'	'	PUNCT
ma-117	369	10	n(0	n(0	PROPN
ma-117	369	11	,	,	PUNCT
ma-117	369	12	1	1	NUM
ma-117	369	13	)	)	PUNCT
ma-117	369	14	(	(	PUNCT
ma-117	369	15	4.8	4.8	NUM
ma-117	369	16	)	)	PUNCT
ma-117	369	17	4.4.1	4.4.1	X
ma-117	369	18	.	.	PUNCT
ma-117	370	1	mean	mean	VERB
ma-117	370	2	stability	stability	NOUN
ma-117	370	3	of	of	ADP
ma-117	370	4	implicit	implicit	ADJ
ma-117	370	5	euler	euler	NOUN
ma-117	370	6	-	-	PUNCT
ma-117	370	7	maruyama	maruyama	NOUN
ma-117	370	8	scheme	scheme	NOUN
ma-117	370	9	.	.	PUNCT
ma-117	371	1	theorem	theorem	VERB
ma-117	371	2	4.5	4.5	NUM
ma-117	371	3	.	.	PUNCT
ma-117	372	1	(	(	PUNCT
ma-117	372	2	mean	mean	VERB
ma-117	372	3	stability	stability	NOUN
ma-117	372	4	of	of	ADP
ma-117	372	5	implicit	implicit	ADJ
ma-117	372	6	euler	euler	NOUN
ma-117	372	7	-	-	PUNCT
ma-117	372	8	maruyama	maruyama	NOUN
ma-117	372	9	scheme	scheme	NOUN
ma-117	372	10	)	)	PUNCT
ma-117	372	11	the	the	DET
ma-117	372	12	implicit	implicit	ADJ
ma-117	372	13	euler	euler	VERB
ma-117	372	14	-	-	PUNCT
ma-117	372	15	maruyama	maruyama	NOUN
ma-117	372	16	scheme	scheme	NOUN
ma-117	372	17	(	(	PUNCT
ma-117	372	18	iem	iem	PROPN
ma-117	372	19	)	)	PUNCT
ma-117	372	20	(	(	PUNCT
ma-117	372	21	4.8	4.8	NUM
ma-117	372	22	)	)	PUNCT
ma-117	372	23	associated	associate	VERB
ma-117	372	24	to	to	ADP
ma-117	372	25	(	(	PUNCT
ma-117	372	26	4.1	4.1	NUM
ma-117	372	27	)	)	PUNCT
ma-117	372	28	model	model	NOUN
ma-117	372	29	is	be	AUX
ma-117	372	30	mean	mean	ADV
ma-117	372	31	asymptotically	asymptotically	ADV
ma-117	372	32	stable	stable	ADJ
ma-117	372	33	if	if	SCONJ
ma-117	372	34	e	e	X
ma-117	372	35	(	(	PUNCT
ma-117	372	36	xiemk+1	xiemk+1	X
ma-117	372	37	)	)	PUNCT
ma-117	373	1	=	=	PUNCT
ma-117	373	2	(	(	PUNCT
ma-117	373	3	1	1	NUM
ma-117	373	4	1−	1−	NUM
ma-117	373	5	θ1∆t	θ1∆t	NOUN
ma-117	373	6	)	)	PUNCT
ma-117	373	7	k+1	k+1	X
ma-117	373	8	e	e	X
ma-117	373	9	(	(	PUNCT
ma-117	373	10	x0	x0	PROPN
ma-117	373	11	)	)	PUNCT
ma-117	373	12	(	(	PUNCT
ma-117	373	13	4.9	4.9	NUM
ma-117	373	14	)	)	PUNCT
ma-117	373	15	with	with	ADP
ma-117	373	16	|1−	|1−	VERB
ma-117	373	17	θ1∆t|	θ1∆t|	PROPN
ma-117	373	18	>	>	X
ma-117	373	19	1	1	NUM
ma-117	373	20	then	then	ADV
ma-117	373	21	,	,	PUNCT
ma-117	373	22	lim	lim	PROPN
ma-117	373	23	∆t→0	∆t→0	PROPN
ma-117	373	24	(	(	PUNCT
ma-117	373	25	lim	lim	PROPN
ma-117	373	26	k→∞	k→∞	PROPN
ma-117	373	27	e	e	PROPN
ma-117	373	28	(	(	PUNCT
ma-117	373	29	∣∣xiemk+1	∣∣xiemk+1	VERB
ma-117	373	30	∣∣2	∣∣2	NUM
ma-117	373	31	)	)	PUNCT
ma-117	373	32	)	)	PUNCT
ma-117	374	1	=	=	SYM
ma-117	374	2	0	0	NUM
ma-117	374	3	proof	proof	NOUN
ma-117	374	4	.	.	PUNCT
ma-117	375	1	let	let	VERB
ma-117	375	2	’s	’s	PRON
ma-117	375	3	evaluate	evaluate	VERB
ma-117	375	4	the	the	DET
ma-117	375	5	mean	mean	NOUN
ma-117	375	6	associated	associate	VERB
ma-117	375	7	to	to	ADP
ma-117	375	8	the	the	DET
ma-117	375	9	implicit	implicit	ADJ
ma-117	375	10	euler	euler	VERB
ma-117	375	11	-	-	PUNCT
ma-117	375	12	maruyama	maruyama	NOUN
ma-117	375	13	scheme	scheme	NOUN
ma-117	375	14	e	e	X
ma-117	375	15	(	(	PUNCT
ma-117	375	16	xiemk+1	xiemk+1	X
ma-117	375	17	)	)	PUNCT
ma-117	376	1	=	=	SYM
ma-117	376	2	e	e	X
ma-117	376	3	(	(	PUNCT
ma-117	376	4	1	1	NUM
ma-117	376	5	1−	1−	NUM
ma-117	376	6	θ1∆t	θ1∆t	NOUN
ma-117	376	7	xk	xk	PROPN
ma-117	377	1	+	+	CCONJ
ma-117	377	2	θ2	θ2	PROPN
ma-117	377	3	1−	1−	NUM
ma-117	377	4	θ1∆t	θ1∆t	NOUN
ma-117	377	5	xk	xk	PROPN
ma-117	377	6	√	√	PROPN
ma-117	377	7	∆tzk	∆tzk	NOUN
ma-117	377	8	)	)	PUNCT
ma-117	378	1	=	=	PUNCT
ma-117	378	2	e	e	X
ma-117	378	3	(	(	PUNCT
ma-117	378	4	1	1	NUM
ma-117	378	5	1−	1−	NUM
ma-117	378	6	θ1∆t	θ1∆t	NOUN
ma-117	378	7	xk	xk	PROPN
ma-117	378	8	)	)	PUNCT
ma-117	379	1	+	+	CCONJ
ma-117	379	2	e	e	X
ma-117	379	3	(	(	PUNCT
ma-117	379	4	θ2	θ2	ADV
ma-117	379	5	1−	1−	NUM
ma-117	379	6	θ1∆t	θ1∆t	NOUN
ma-117	379	7	√	√	NUM
ma-117	379	8	∆t	∆t	PROPN
ma-117	379	9	)	)	PUNCT
ma-117	379	10	(	(	PUNCT
ma-117	379	11	xk	xk	PROPN
ma-117	379	12	)	)	PUNCT
ma-117	379	13	(	(	PUNCT
ma-117	379	14	zk	zk	PROPN
ma-117	379	15	)	)	PUNCT
ma-117	379	16	=	=	SYM
ma-117	379	17	e	e	X
ma-117	379	18	(	(	PUNCT
ma-117	379	19	1	1	NUM
ma-117	379	20	1−	1−	NUM
ma-117	379	21	θ1∆t	θ1∆t	NOUN
ma-117	379	22	xk	xk	PROPN
ma-117	379	23	)	)	PUNCT
ma-117	380	1	=	=	PUNCT
ma-117	380	2	1	1	NUM
ma-117	380	3	1−	1−	NUM
ma-117	380	4	θ1∆t	θ1∆t	NOUN
ma-117	380	5	e	e	X
ma-117	380	6	(	(	PUNCT
ma-117	380	7	xk	xk	PROPN
ma-117	380	8	)	)	PUNCT
ma-117	380	9	=	=	SYM
ma-117	381	1	(	(	PUNCT
ma-117	381	2	1	1	NUM
ma-117	381	3	1−	1−	NUM
ma-117	381	4	θ1∆t	θ1∆t	NOUN
ma-117	381	5	)	)	PUNCT
ma-117	381	6	2	2	NUM
ma-117	381	7	e	e	X
ma-117	381	8	(	(	PUNCT
ma-117	381	9	xk−1	xk−1	PROPN
ma-117	381	10	)	)	PUNCT
ma-117	381	11	=	=	PRON
ma-117	382	1	(	(	PUNCT
ma-117	382	2	1	1	NUM
ma-117	382	3	1−	1−	NUM
ma-117	382	4	θ1∆t	θ1∆t	NOUN
ma-117	382	5	)	)	PUNCT
ma-117	382	6	3	3	NUM
ma-117	382	7	e	e	X
ma-117	382	8	(	(	PUNCT
ma-117	382	9	xk−2	xk−2	PROPN
ma-117	382	10	)	)	PUNCT
ma-117	382	11	...	...	PUNCT
ma-117	383	1	=	=	PUNCT
ma-117	383	2	(	(	PUNCT
ma-117	383	3	1	1	NUM
ma-117	383	4	1−	1−	NUM
ma-117	383	5	θ1∆t	θ1∆t	NOUN
ma-117	383	6	)	)	PUNCT
ma-117	383	7	k+1	k+1	X
ma-117	384	1	e	e	X
ma-117	384	2	(	(	PUNCT
ma-117	384	3	x0	x0	PROPN
ma-117	384	4	)	)	PUNCT
ma-117	384	5	continuing	continue	VERB
ma-117	384	6	with	with	ADP
ma-117	384	7	the	the	DET
ma-117	384	8	iterations	iteration	NOUN
ma-117	384	9	we	we	PRON
ma-117	384	10	get	get	VERB
ma-117	384	11	:	:	PUNCT
ma-117	384	12	e	e	X
ma-117	384	13	(	(	PUNCT
ma-117	384	14	xiemk+1	xiemk+1	X
ma-117	384	15	)	)	PUNCT
ma-117	384	16	=	=	PUNCT
ma-117	384	17	(	(	PUNCT
ma-117	384	18	1	1	NUM
ma-117	384	19	1−	1−	NUM
ma-117	384	20	θ1∆t	θ1∆t	NOUN
ma-117	384	21	)	)	PUNCT
ma-117	384	22	k+1	k+1	X
ma-117	384	23	e	e	X
ma-117	384	24	(	(	PUNCT
ma-117	384	25	x0	x0	PROPN
ma-117	384	26	)	)	PUNCT
ma-117	384	27	https://doi.org/10.28924/ada/ma.3.8	https://doi.org/10.28924/ada/ma.3.8	NUM
ma-117	384	28	eur	eur	NOUN
ma-117	384	29	.	.	PUNCT
ma-117	385	1	j.	j.	PROPN
ma-117	385	2	math	math	PROPN
ma-117	385	3	.	.	PUNCT
ma-117	386	1	anal	anal	PROPN
ma-117	386	2	.	.	PUNCT
ma-117	387	1	10.28924	10.28924	NUM
ma-117	387	2	/	/	SYM
ma-117	387	3	ada	ada	PROPN
ma-117	387	4	/	/	SYM
ma-117	388	1	ma.3.8	ma.3.8	PROPN
ma-117	388	2	17passing	17passe	VERB
ma-117	388	3	to	to	ADP
ma-117	388	4	the	the	DET
ma-117	388	5	limit	limit	NOUN
ma-117	388	6	with	with	ADP
ma-117	388	7	∆t	∆t	PROPN
ma-117	388	8	→	→	SYM
ma-117	388	9	0	0	NUM
ma-117	388	10	and	and	CCONJ
ma-117	388	11	k	k	PROPN
ma-117	388	12	→	→	SYM
ma-117	388	13	+	+	NOUN
ma-117	388	14	∞	∞	PROPN
ma-117	388	15	,	,	PUNCT
ma-117	388	16	we	we	PRON
ma-117	388	17	obtain	obtain	VERB
ma-117	388	18	the	the	DET
ma-117	388	19	stated	state	VERB
ma-117	388	20	results	result	NOUN
ma-117	388	21	,	,	PUNCT
ma-117	388	22	ie	ie	ADV
ma-117	388	23	lim	lim	PROPN
ma-117	388	24	∆t→0	∆t→0	PROPN
ma-117	388	25	(	(	PUNCT
ma-117	388	26	lim	lim	PROPN
ma-117	388	27	k→∞	k→∞	PROPN
ma-117	388	28	e	e	PROPN
ma-117	388	29	(	(	PUNCT
ma-117	388	30	∣∣xiemk+1	∣∣xiemk+1	VERB
ma-117	388	31	∣∣2	∣∣2	NUM
ma-117	388	32	)	)	PUNCT
ma-117	388	33	)	)	PUNCT
ma-117	389	1	=	=	SYM
ma-117	389	2	0	0	NUM
ma-117	389	3	�	�	PROPN
ma-117	389	4	4.4.2	4.4.2	NUM
ma-117	389	5	.	.	PUNCT
ma-117	390	1	mean	mean	ADJ
ma-117	390	2	-	-	PUNCT
ma-117	390	3	square	square	ADJ
ma-117	390	4	stability	stability	NOUN
ma-117	390	5	of	of	ADP
ma-117	390	6	implicit	implicit	ADJ
ma-117	390	7	euler	euler	NOUN
ma-117	390	8	-	-	PUNCT
ma-117	390	9	maruyama	maruyama	NOUN
ma-117	390	10	scheme	scheme	NOUN
ma-117	390	11	.	.	PUNCT
ma-117	391	1	theorem	theorem	VERB
ma-117	391	2	4.6	4.6	NUM
ma-117	391	3	.	.	PUNCT
ma-117	392	1	(	(	PUNCT
ma-117	392	2	mean	mean	ADJ
ma-117	392	3	-	-	PUNCT
ma-117	392	4	square	square	ADJ
ma-117	392	5	stability	stability	NOUN
ma-117	392	6	of	of	ADP
ma-117	392	7	implicit	implicit	ADJ
ma-117	392	8	euler	euler	NOUN
ma-117	392	9	-	-	PUNCT
ma-117	392	10	maruyama	maruyama	NOUN
ma-117	392	11	scheme	scheme	NOUN
ma-117	392	12	)	)	PUNCT
ma-117	392	13	the	the	DET
ma-117	392	14	implicit	implicit	ADJ
ma-117	392	15	eulermaruyama	eulermaruyama	NOUN
ma-117	392	16	scheme	scheme	NOUN
ma-117	392	17	associated	associate	VERB
ma-117	392	18	to	to	ADP
ma-117	392	19	the	the	DET
ma-117	392	20	model	model	NOUN
ma-117	392	21	(	(	PUNCT
ma-117	392	22	4.1	4.1	NUM
ma-117	392	23	)	)	PUNCT
ma-117	392	24	is	be	AUX
ma-117	392	25	asymptotically	asymptotically	ADV
ma-117	392	26	mean	mean	ADJ
ma-117	392	27	-	-	PUNCT
ma-117	392	28	square	square	ADJ
ma-117	392	29	stable	stable	NOUN
ma-117	392	30	if	if	SCONJ
ma-117	392	31	e	e	PROPN
ma-117	392	32	(	(	PUNCT
ma-117	392	33	∣∣xiemk+1	∣∣xiemk+1	X
ma-117	392	34	∣∣2	∣∣2	NUM
ma-117	392	35	)	)	PUNCT
ma-117	392	36	=	=	NOUN
ma-117	393	1	[	[	PUNCT
ma-117	393	2	1	1	NUM
ma-117	393	3	+	+	CCONJ
ma-117	393	4	∣∣θ2	∣∣θ2	X
ma-117	393	5	√	√	NUM
ma-117	393	6	∆t	∆t	PROPN
ma-117	393	7	∣∣	∣∣	NUM
ma-117	393	8	1−	1−	NUM
ma-117	393	9	θ1∆t	θ1∆t	NOUN
ma-117	393	10	]	]	PUNCT
ma-117	393	11	2(k+1	2(k+1	NUM
ma-117	393	12	)	)	PUNCT
ma-117	393	13	e	e	NOUN
ma-117	393	14	(	(	PUNCT
ma-117	393	15	|x0|2	|x0|2	PROPN
ma-117	393	16	)	)	PUNCT
ma-117	393	17	with	with	ADP
ma-117	393	18	∣∣∣∣1+|θ2	∣∣∣∣1+|θ2	NOUN
ma-117	393	19	√	√	ADP
ma-117	393	20	∆t|	∆t|	PROPN
ma-117	393	21	1−θ1∆t	1−θ1∆t	PROPN
ma-117	393	22	∣∣∣∣	∣∣∣∣	NOUN
ma-117	393	23	<	<	X
ma-117	393	24	1	1	NUM
ma-117	393	25	and	and	CCONJ
ma-117	393	26	lim	lim	PROPN
ma-117	393	27	∆t→0	∆t→0	PROPN
ma-117	393	28	(	(	PUNCT
ma-117	393	29	lim	lim	PROPN
ma-117	393	30	k→∞	k→∞	PROPN
ma-117	393	31	e	e	PROPN
ma-117	393	32	(	(	PUNCT
ma-117	393	33	∣∣xemik+1	∣∣xemik+1	NOUN
ma-117	393	34	∣∣2	∣∣2	NUM
ma-117	393	35	)	)	PUNCT
ma-117	393	36	)	)	PUNCT
ma-117	394	1	=	=	SYM
ma-117	394	2	0	0	NUM
ma-117	394	3	proof	proof	NOUN
ma-117	394	4	.	.	PUNCT
ma-117	395	1	:	:	PUNCT
ma-117	395	2	let	let	VERB
ma-117	395	3	us	we	PRON
ma-117	395	4	calculate	calculate	VERB
ma-117	395	5	the	the	DET
ma-117	395	6	quadratic	quadratic	ADJ
ma-117	395	7	mean	mean	NOUN
ma-117	395	8	,	,	PUNCT
ma-117	395	9	in	in	ADP
ma-117	395	10	effect	effect	NOUN
ma-117	395	11	,	,	PUNCT
ma-117	395	12	e	e	X
ma-117	395	13	(	(	PUNCT
ma-117	395	14	∣∣xmk+1	∣∣xmk+1	X
ma-117	395	15	∣∣2	∣∣2	NUM
ma-117	395	16	)	)	PUNCT
ma-117	395	17	=	=	PUNCT
ma-117	396	1	e	e	NOUN
ma-117	396	2	∣∣∣∣∣	∣∣∣∣∣	NUM
ma-117	396	3	1	1	NUM
ma-117	396	4	1−	1−	NUM
ma-117	396	5	θ1∆t	θ1∆t	NOUN
ma-117	396	6	xt	xt	PUNCT
ma-117	397	1	+	+	CCONJ
ma-117	397	2	θ2	θ2	ADV
ma-117	397	3	√	√	ADV
ma-117	397	4	∆t	∆t	PROPN
ma-117	397	5	1−	1−	NUM
ma-117	397	6	θ1∆t	θ1∆t	NOUN
ma-117	397	7	xtzt	xtzt	NOUN
ma-117	398	1	∣∣∣∣∣	∣∣∣∣∣	ADP
ma-117	398	2	2	2	NUM
ma-117	398	3			PROPN
ma-117	398	4	=	=	PUNCT
ma-117	398	5	(	(	PUNCT
ma-117	398	6	1	1	NUM
ma-117	398	7	1−	1−	NUM
ma-117	398	8	θ1∆t	θ1∆t	NOUN
ma-117	398	9	)	)	PUNCT
ma-117	398	10	2	2	NUM
ma-117	398	11	e	e	NOUN
ma-117	398	12	(	(	PUNCT
ma-117	398	13	∣∣∣xk	∣∣∣xk	PROPN
ma-117	398	14	+	+	PROPN
ma-117	398	15	θ2	θ2	ADV
ma-117	398	16	√	√	ADV
ma-117	398	17	∆txkzk	∆txkzk	ADJ
ma-117	398	18	∣∣∣2	∣∣∣2	NUM
ma-117	398	19	)	)	PUNCT
ma-117	398	20	=	=	PUNCT
ma-117	398	21	(	(	PUNCT
ma-117	398	22	1	1	NUM
ma-117	398	23	1−	1−	NUM
ma-117	398	24	θ1∆t	θ1∆t	NOUN
ma-117	398	25	)	)	PUNCT
ma-117	398	26	2	2	NUM
ma-117	398	27	[	[	PUNCT
ma-117	398	28	e	e	X
ma-117	398	29	(	(	PUNCT
ma-117	398	30	|xk	|xk	X
ma-117	398	31	|2	|2	NUM
ma-117	398	32	)	)	PUNCT
ma-117	399	1	+	+	CCONJ
ma-117	399	2	e	e	X
ma-117	399	3	(	(	PUNCT
ma-117	399	4	∣∣∣+θ2	∣∣∣+θ2	NOUN
ma-117	399	5	√	√	ADP
ma-117	399	6	∆txkzk	∆txkzk	ADJ
ma-117	399	7	∣∣∣2	∣∣∣2	NUM
ma-117	399	8	)	)	PUNCT
ma-117	399	9	]	]	PUNCT
ma-117	400	1	=	=	PUNCT
ma-117	400	2	∣∣∣∣	∣∣∣∣	NOUN
ma-117	400	3	1	1	NUM
ma-117	400	4	1−	1−	NUM
ma-117	400	5	θ1∆t	θ1∆t	NOUN
ma-117	400	6	∣∣∣∣2	∣∣∣∣2	NOUN
ma-117	401	1	[	[	X
ma-117	401	2	e	e	X
ma-117	401	3	(	(	PUNCT
ma-117	401	4	|xk	|xk	X
ma-117	401	5	|2)+	|2)+	PROPN
ma-117	401	6	∣∣∣+θ2	∣∣∣+θ2	NOUN
ma-117	401	7	√	√	NUM
ma-117	401	8	∆t	∆t	PROPN
ma-117	401	9	∣∣∣2	∣∣∣2	NOUN
ma-117	401	10	e	e	X
ma-117	401	11	(	(	PUNCT
ma-117	401	12	|xk	|xk	X
ma-117	401	13	|2	|2	NUM
ma-117	401	14	)	)	PUNCT
ma-117	401	15	]	]	PUNCT
ma-117	402	1	=	=	PUNCT
ma-117	402	2	(	(	PUNCT
ma-117	402	3	1	1	NUM
ma-117	402	4	+	+	CCONJ
ma-117	402	5	∣∣θ2	∣∣θ2	X
ma-117	402	6	√	√	NUM
ma-117	402	7	∆t	∆t	PROPN
ma-117	402	8	∣∣2	∣∣2	NUM
ma-117	402	9	)	)	PUNCT
ma-117	402	10	(	(	PUNCT
ma-117	402	11	1−	1−	NUM
ma-117	402	12	θ1∆t)2	θ1∆t)2	NOUN
ma-117	402	13	e	e	X
ma-117	402	14	(	(	PUNCT
ma-117	402	15	|xk	|xk	X
ma-117	402	16	|2	|2	NUM
ma-117	402	17	)	)	PUNCT
ma-117	402	18	=	=	PUNCT
ma-117	402	19	(	(	PUNCT
ma-117	402	20	1	1	NUM
ma-117	402	21	+	+	CCONJ
ma-117	402	22	∣∣θ2	∣∣θ2	X
ma-117	402	23	√	√	NUM
ma-117	402	24	∆t	∆t	PROPN
ma-117	402	25	∣∣2	∣∣2	NUM
ma-117	402	26	)	)	PUNCT
ma-117	402	27	(	(	PUNCT
ma-117	402	28	1−	1−	NUM
ma-117	402	29	θ1∆t)2	θ1∆t)2	NOUN
ma-117	402	30	(	(	PUNCT
ma-117	402	31	1	1	NUM
ma-117	402	32	+	+	CCONJ
ma-117	402	33	∣∣θ2	∣∣θ2	X
ma-117	402	34	√	√	NUM
ma-117	402	35	∆t	∆t	PROPN
ma-117	402	36	∣∣2	∣∣2	NUM
ma-117	402	37	)	)	PUNCT
ma-117	402	38	(	(	PUNCT
ma-117	402	39	1−	1−	NUM
ma-117	402	40	θ1∆t)2	θ1∆t)2	NOUN
ma-117	402	41	e	e	X
ma-117	402	42	(	(	PUNCT
ma-117	402	43	|xk−1|2	|xk−1|2	X
ma-117	402	44	)	)	PUNCT
ma-117	402	45	continuing	continue	VERB
ma-117	402	46	with	with	ADP
ma-117	402	47	the	the	DET
ma-117	402	48	iterations	iteration	NOUN
ma-117	402	49	,	,	PUNCT
ma-117	402	50	we	we	PRON
ma-117	402	51	get	get	VERB
ma-117	402	52	:	:	PUNCT
ma-117	402	53	e	e	NOUN
ma-117	402	54	(	(	PUNCT
ma-117	402	55	∣∣xemik+1	∣∣xemik+1	NOUN
ma-117	402	56	∣∣2	∣∣2	NUM
ma-117	402	57	)	)	PUNCT
ma-117	402	58	=	=	PRON
ma-117	403	1	[	[	PUNCT
ma-117	403	2	1	1	NUM
ma-117	403	3	+	+	CCONJ
ma-117	403	4	∣∣θ2	∣∣θ2	X
ma-117	403	5	√	√	NUM
ma-117	403	6	∆t	∆t	PROPN
ma-117	403	7	∣∣	∣∣	NUM
ma-117	403	8	1−	1−	NUM
ma-117	403	9	θ1∆t	θ1∆t	NOUN
ma-117	403	10	]	]	PUNCT
ma-117	403	11	2(k+1	2(k+1	NUM
ma-117	403	12	)	)	PUNCT
ma-117	403	13	e	e	NOUN
ma-117	403	14	(	(	PUNCT
ma-117	403	15	|x0|2	|x0|2	X
ma-117	403	16	)	)	PUNCT
ma-117	403	17	passing	pass	VERB
ma-117	403	18	to	to	ADP
ma-117	403	19	the	the	DET
ma-117	403	20	limit	limit	NOUN
ma-117	403	21	with	with	ADP
ma-117	403	22	∆t	∆t	PROPN
ma-117	403	23	→	→	SYM
ma-117	403	24	0	0	NUM
ma-117	403	25	and	and	CCONJ
ma-117	403	26	k	k	PROPN
ma-117	403	27	→	→	SYM
ma-117	403	28	+	+	NOUN
ma-117	403	29	∞	∞	PROPN
ma-117	403	30	,	,	PUNCT
ma-117	403	31	we	we	PRON
ma-117	403	32	obtain	obtain	VERB
ma-117	403	33	the	the	DET
ma-117	403	34	stated	state	VERB
ma-117	403	35	results	result	NOUN
ma-117	403	36	,	,	PUNCT
ma-117	403	37	i.e	i.e	PROPN
ma-117	403	38	:	:	PUNCT
ma-117	403	39	lim	lim	PROPN
ma-117	403	40	∆t→0	∆t→0	PROPN
ma-117	403	41	(	(	PUNCT
ma-117	403	42	lim	lim	PROPN
ma-117	403	43	k→∞	k→∞	PROPN
ma-117	403	44	e	e	PROPN
ma-117	403	45	(	(	PUNCT
ma-117	403	46	∣∣xmk+1	∣∣xmk+1	X
ma-117	403	47	∣∣2	∣∣2	NUM
ma-117	403	48	)	)	PUNCT
ma-117	403	49	)	)	PUNCT
ma-117	404	1	=	=	SYM
ma-117	404	2	0	0	NUM
ma-117	404	3	�	�	PROPN
ma-117	404	4	https://doi.org/10.28924/ada/ma.3.8	https://doi.org/10.28924/ada/ma.3.8	PROPN
ma-117	404	5	eur	eur	PROPN
ma-117	404	6	.	.	PUNCT
ma-117	405	1	j.	j.	PROPN
ma-117	405	2	math	math	PROPN
ma-117	405	3	.	.	PUNCT
ma-117	406	1	anal	anal	PROPN
ma-117	406	2	.	.	PUNCT
ma-117	407	1	10.28924	10.28924	NUM
ma-117	407	2	/	/	SYM
ma-117	407	3	ada	ada	PROPN
ma-117	407	4	/	/	SYM
ma-117	407	5	ma.3.8	ma.3.8	PROPN
ma-117	407	6	185	185	NUM
ma-117	407	7	.	.	PUNCT
ma-117	408	1	numerical	numerical	ADJ
ma-117	408	2	simulations	simulation	NOUN
ma-117	408	3	and	and	CCONJ
ma-117	408	4	residual	residual	ADJ
ma-117	408	5	calculations	calculation	NOUN
ma-117	408	6	in	in	ADP
ma-117	408	7	this	this	DET
ma-117	408	8	section	section	NOUN
ma-117	408	9	,	,	PUNCT
ma-117	408	10	we	we	PRON
ma-117	408	11	present	present	VERB
ma-117	408	12	some	some	DET
ma-117	408	13	numerical	numerical	ADJ
ma-117	408	14	simulations	simulation	NOUN
ma-117	408	15	for	for	ADP
ma-117	408	16	vasicek	vasicek	NOUN
ma-117	408	17	and	and	CCONJ
ma-117	408	18	geometric	geometric	ADJ
ma-117	408	19	brownianmotion	brownianmotion	NOUN
ma-117	408	20	models	model	NOUN
ma-117	408	21	using	use	VERB
ma-117	408	22	matlab	matlab	PROPN
ma-117	408	23	and	and	CCONJ
ma-117	408	24	we	we	PRON
ma-117	408	25	calculate	calculate	VERB
ma-117	408	26	the	the	DET
ma-117	408	27	errors	error	NOUN
ma-117	408	28	between	between	ADP
ma-117	408	29	the	the	DET
ma-117	408	30	exact	exact	ADJ
ma-117	408	31	solution	solution	NOUN
ma-117	408	32	and	and	CCONJ
ma-117	408	33	thatobtained	thatobtaine	VERB
ma-117	408	34	by	by	ADP
ma-117	408	35	applying	apply	VERB
ma-117	408	36	the	the	DET
ma-117	408	37	numerical	numerical	ADJ
ma-117	408	38	schemes	scheme	NOUN
ma-117	408	39	of	of	ADP
ma-117	408	40	euler	euler	NOUN
ma-117	408	41	-	-	PUNCT
ma-117	408	42	maruyama	maruyama	NOUN
ma-117	408	43	,	,	PUNCT
ma-117	408	44	milshtein	milshtein	NOUN
ma-117	408	45	and	and	CCONJ
ma-117	408	46	implicit	implicit	ADJ
ma-117	408	47	euler	euler	NOUN
ma-117	408	48	-	-	PUNCT
ma-117	408	49	maruyama	maruyama	NOUN
ma-117	408	50	.	.	PUNCT
ma-117	409	1	5.1	5.1	NUM
ma-117	409	2	.	.	PUNCT
ma-117	410	1	numerical	numerical	PROPN
ma-117	410	2	simulation	simulation	PROPN
ma-117	410	3	of	of	ADP
ma-117	410	4	vasicek	vasicek	PROPN
ma-117	410	5	and	and	CCONJ
ma-117	410	6	geometric	geometric	ADJ
ma-117	410	7	brownian	brownian	ADJ
ma-117	410	8	motion	motion	NOUN
ma-117	410	9	models	model	NOUN
ma-117	410	10	.	.	PUNCT
ma-117	411	1	we	we	PRON
ma-117	411	2	present	present	VERB
ma-117	411	3	somesimulations	somesimulation	NOUN
ma-117	411	4	of	of	ADP
ma-117	411	5	vasicek	vasicek	PROPN
ma-117	411	6	and	and	CCONJ
ma-117	411	7	brownian	brownian	ADJ
ma-117	411	8	geometric	geometric	ADJ
ma-117	411	9	motion	motion	NOUN
ma-117	411	10	models	model	NOUN
ma-117	411	11	in	in	ADP
ma-117	411	12	the	the	DET
ma-117	411	13	increasing	increase	VERB
ma-117	411	14	and	and	CCONJ
ma-117	411	15	decreasingcases	decreasingcase	NOUN
ma-117	411	16	.	.	PUNCT
ma-117	412	1	figure	figure	NOUN
ma-117	412	2	1	1	NUM
ma-117	412	3	.	.	PUNCT
ma-117	412	4	increasing	increase	VERB
ma-117	412	5	vasicek	vasicek	PROPN
ma-117	412	6	model	model	NOUN
ma-117	412	7	https://doi.org/10.28924/ada/ma.3.8	https://doi.org/10.28924/ada/ma.3.8	PROPN
ma-117	412	8	eur	eur	PROPN
ma-117	412	9	.	.	PUNCT
ma-117	413	1	j.	j.	PROPN
ma-117	413	2	math	math	PROPN
ma-117	413	3	.	.	PUNCT
ma-117	414	1	anal	anal	PROPN
ma-117	414	2	.	.	PUNCT
ma-117	415	1	10.28924	10.28924	NUM
ma-117	415	2	/	/	SYM
ma-117	415	3	ada	ada	PROPN
ma-117	415	4	/	/	SYM
ma-117	415	5	ma.3.8	ma.3.8	PROPN
ma-117	415	6	19	19	NUM
ma-117	415	7	figure	figure	NOUN
ma-117	415	8	2	2	NUM
ma-117	415	9	.	.	PUNCT
ma-117	415	10	decreasing	decrease	VERB
ma-117	415	11	vasicek	vasicek	PROPN
ma-117	415	12	model	model	NOUN
ma-117	415	13	figure	figure	NOUN
ma-117	415	14	3	3	NUM
ma-117	415	15	.	.	PUNCT
ma-117	415	16	increasing	increase	VERB
ma-117	415	17	geometric	geometric	ADJ
ma-117	415	18	brownian	brownian	ADJ
ma-117	415	19	motion	motion	NOUN
ma-117	415	20	model	model	NOUN
ma-117	415	21	https://doi.org/10.28924/ada/ma.3.8	https://doi.org/10.28924/ada/ma.3.8	PROPN
ma-117	415	22	eur	eur	PROPN
ma-117	415	23	.	.	PUNCT
ma-117	416	1	j.	j.	PROPN
ma-117	416	2	math	math	PROPN
ma-117	416	3	.	.	PUNCT
ma-117	417	1	anal	anal	PROPN
ma-117	417	2	.	.	PUNCT
ma-117	418	1	10.28924	10.28924	NUM
ma-117	418	2	/	/	SYM
ma-117	418	3	ada	ada	PROPN
ma-117	418	4	/	/	SYM
ma-117	418	5	ma.3.8	ma.3.8	PROPN
ma-117	418	6	20	20	NUM
ma-117	418	7	figure	figure	NOUN
ma-117	418	8	4	4	NUM
ma-117	418	9	.	.	PUNCT
ma-117	418	10	decreasing	decrease	VERB
ma-117	418	11	geometric	geometric	ADJ
ma-117	418	12	brownian	brownian	ADJ
ma-117	418	13	motion	motion	NOUN
ma-117	418	14	model	model	NOUN
ma-117	418	15	5.2	5.2	NUM
ma-117	418	16	.	.	PUNCT
ma-117	419	1	interpretation	interpretation	NOUN
ma-117	419	2	of	of	ADP
ma-117	419	3	results	result	NOUN
ma-117	419	4	.	.	PUNCT
ma-117	420	1	5.2.1	5.2.1	X
ma-117	420	2	.	.	PUNCT
ma-117	420	3	vasicek	vasicek	PROPN
ma-117	420	4	model	model	NOUN
ma-117	420	5	.	.	PUNCT
ma-117	421	1	the	the	DET
ma-117	421	2	figures	figure	NOUN
ma-117	421	3	1	1	NUM
ma-117	421	4	and	and	CCONJ
ma-117	421	5	2	2	NUM
ma-117	421	6	show	show	VERB
ma-117	421	7	the	the	DET
ma-117	421	8	stability	stability	NOUN
ma-117	421	9	of	of	ADP
ma-117	421	10	the	the	DET
ma-117	421	11	vasicek	vasicek	PROPN
ma-117	421	12	model	model	NOUN
ma-117	421	13	in	in	ADP
ma-117	421	14	the	the	DET
ma-117	421	15	increasingand	increasingand	NOUN
ma-117	421	16	decreasing	decrease	VERB
ma-117	421	17	cases	case	NOUN
ma-117	421	18	,	,	PUNCT
ma-117	421	19	in	in	ADP
ma-117	421	20	these	these	DET
ma-117	421	21	figures	figure	NOUN
ma-117	421	22	we	we	PRON
ma-117	421	23	see	see	VERB
ma-117	421	24	that	that	SCONJ
ma-117	421	25	the	the	DET
ma-117	421	26	euler	euler	PROPN
ma-117	421	27	-	-	PUNCT
ma-117	421	28	maruyama	maruyama	NOUN
ma-117	421	29	scheme	scheme	NOUN
ma-117	421	30	coincides	coincide	NOUN
ma-117	421	31	with	with	ADP
ma-117	421	32	thatof	thatof	PROPN
ma-117	421	33	milshtein	milshtein	PROPN
ma-117	421	34	.	.	PUNCT
ma-117	422	1	we	we	PRON
ma-117	422	2	have	have	VERB
ma-117	422	3	in	in	ADP
ma-117	422	4	the	the	DET
ma-117	422	5	first	first	ADJ
ma-117	422	6	two	two	NUM
ma-117	422	7	figures	figure	NOUN
ma-117	422	8	of	of	ADP
ma-117	422	9	figure	figure	NOUN
ma-117	422	10	1	1	NUM
ma-117	422	11	the	the	DET
ma-117	422	12	following	follow	VERB
ma-117	422	13	errors	error	NOUN
ma-117	422	14	:	:	PUNCT
ma-117	422	15	emerr	emerr	NOUN
ma-117	422	16	=	=	NOUN
ma-117	422	17	0.2280,milerr	0.2280,milerr	NUM
ma-117	423	1	=	=	PUNCT
ma-117	423	2	0.2280	0.2280	NUM
ma-117	423	3	and	and	CCONJ
ma-117	423	4	iemerr	iemerr	NOUN
ma-117	423	5	=	=	SYM
ma-117	423	6	0.2258	0.2258	NUM
ma-117	423	7	in	in	ADP
ma-117	423	8	both	both	DET
ma-117	423	9	figures	figure	NOUN
ma-117	423	10	of	of	ADP
ma-117	423	11	figure	figure	NOUN
ma-117	423	12	1	1	NUM
ma-117	423	13	emerr	emerr	NOUN
ma-117	423	14	=	=	PUNCT
ma-117	423	15	0.3007	0.3007	NUM
ma-117	423	16	,	,	PUNCT
ma-117	423	17	milerr	milerr	NOUN
ma-117	423	18	=	=	SYM
ma-117	423	19	0.3007and	0.3007and	NUM
ma-117	423	20	iemerr	iemerr	NOUN
ma-117	423	21	=	=	SYM
ma-117	423	22	0.2851	0.2851	NUM
ma-117	423	23	in	in	ADP
ma-117	423	24	both	both	DET
ma-117	423	25	figures	figure	NOUN
ma-117	423	26	of	of	ADP
ma-117	423	27	figure	figure	NOUN
ma-117	423	28	2	2	NUM
ma-117	423	29	emerr	emerr	NOUN
ma-117	423	30	=	=	PUNCT
ma-117	423	31	0.2268	0.2268	NUM
ma-117	423	32	,	,	PUNCT
ma-117	423	33	milerr	milerr	NOUN
ma-117	423	34	=	=	SYM
ma-117	423	35	0.2268	0.2268	NUM
ma-117	423	36	and	and	CCONJ
ma-117	423	37	iemerr	iemerr	NOUN
ma-117	423	38	=	=	SYM
ma-117	423	39	0.2237	0.2237	NUM
ma-117	423	40	.	.	PUNCT
ma-117	424	1	5.2.2	5.2.2	NUM
ma-117	424	2	.	.	PUNCT
ma-117	424	3	geometric	geometric	ADJ
ma-117	424	4	brownian	brownian	ADJ
ma-117	424	5	motion	motion	NOUN
ma-117	424	6	model	model	NOUN
ma-117	424	7	.	.	PUNCT
ma-117	425	1	the	the	DET
ma-117	425	2	figures	figure	NOUN
ma-117	425	3	figure	figure	VERB
ma-117	425	4	3	3	NUM
ma-117	425	5	et	et	NOUN
ma-117	425	6	figure	figure	VERB
ma-117	425	7	4	4	NUM
ma-117	425	8	present	present	ADJ
ma-117	425	9	the	the	DET
ma-117	425	10	stability	stability	NOUN
ma-117	425	11	ofgeometric	ofgeometric	ADJ
ma-117	425	12	brownian	brownian	ADJ
ma-117	425	13	motion	motion	NOUN
ma-117	425	14	in	in	ADP
ma-117	425	15	the	the	DET
ma-117	425	16	increasing	increase	VERB
ma-117	425	17	and	and	CCONJ
ma-117	425	18	decreasing	decrease	VERB
ma-117	425	19	cases	case	NOUN
ma-117	425	20	.	.	PUNCT
ma-117	426	1	indeed	indeed	ADV
ma-117	426	2	,	,	PUNCT
ma-117	426	3	the	the	DET
ma-117	426	4	first	first	ADJ
ma-117	426	5	three	three	NUM
ma-117	426	6	figuresin	figuresin	NOUN
ma-117	426	7	figure	figure	NOUN
ma-117	426	8	3	3	NUM
ma-117	426	9	present	present	VERB
ma-117	426	10	the	the	DET
ma-117	426	11	increasing	increase	VERB
ma-117	426	12	stability	stability	NOUN
ma-117	426	13	of	of	ADP
ma-117	426	14	geometric	geometric	ADJ
ma-117	426	15	motion	motion	NOUN
ma-117	426	16	and	and	CCONJ
ma-117	426	17	the	the	DET
ma-117	426	18	last	last	ADJ
ma-117	426	19	figure	figure	NOUN
ma-117	426	20	in	in	ADP
ma-117	426	21	figure	figure	NOUN
ma-117	426	22	3	3	NUM
ma-117	426	23	andthe	andthe	PROPN
ma-117	426	24	two	two	NUM
ma-117	426	25	figures	figure	NOUN
ma-117	426	26	in	in	ADP
ma-117	426	27	figure	figure	NOUN
ma-117	426	28	4	4	NUM
ma-117	426	29	show	show	VERB
ma-117	426	30	the	the	DET
ma-117	426	31	decreasing	decrease	VERB
ma-117	426	32	stability	stability	NOUN
ma-117	426	33	of	of	ADP
ma-117	426	34	the	the	DET
ma-117	426	35	model	model	NOUN
ma-117	426	36	.	.	PUNCT
ma-117	427	1	we	we	PRON
ma-117	427	2	have	have	VERB
ma-117	427	3	in	in	ADP
ma-117	427	4	the	the	DET
ma-117	427	5	first	first	ADJ
ma-117	427	6	twofigures	twofigure	NOUN
ma-117	427	7	and	and	CCONJ
ma-117	427	8	figure	figure	VERB
ma-117	427	9	3	3	NUM
ma-117	427	10	the	the	DET
ma-117	427	11	following	follow	VERB
ma-117	427	12	errors	error	NOUN
ma-117	427	13	:	:	PUNCT
ma-117	427	14	emerr	emerr	NOUN
ma-117	427	15	=	=	NOUN
ma-117	427	16	0.0027	0.0027	NUM
ma-117	427	17	,	,	PUNCT
ma-117	427	18	milerr	milerr	NOUN
ma-117	427	19	=	=	SYM
ma-117	427	20	0.0011	0.0011	NUM
ma-117	427	21	and	and	CCONJ
ma-117	427	22	iemerr	iemerr	NOUN
ma-117	427	23	=	=	SYM
ma-117	427	24	0.0013and	0.0013and	NOUN
ma-117	427	25	for	for	ADP
ma-117	427	26	the	the	DET
ma-117	427	27	third	third	ADJ
ma-117	427	28	figure	figure	NOUN
ma-117	427	29	in	in	ADP
ma-117	427	30	figure	figure	NOUN
ma-117	427	31	3	3	NUM
ma-117	427	32	:	:	PUNCT
ma-117	427	33	emerr	emerr	NOUN
ma-117	427	34	=	=	SYM
ma-117	427	35	0.0177	0.0177	NUM
ma-117	427	36	,	,	PUNCT
ma-117	427	37	milerr	milerr	NOUN
ma-117	427	38	=	=	SYM
ma-117	427	39	0.0111	0.0111	NUM
ma-117	427	40	and	and	CCONJ
ma-117	427	41	iemerr	iemerr	NOUN
ma-117	427	42	=	=	NOUN
ma-117	427	43	0.0128	0.0128	NUM
ma-117	427	44	.	.	PUNCT
ma-117	428	1	in	in	ADP
ma-117	428	2	theboth	theboth	NOUN
ma-117	428	3	figures	figure	NOUN
ma-117	428	4	of	of	ADP
ma-117	428	5	figure	figure	NOUN
ma-117	428	6	4	4	NUM
ma-117	428	7	:	:	PUNCT
ma-117	428	8	emerr	emerr	NOUN
ma-117	428	9	=	=	SYM
ma-117	428	10	0.0054	0.0054	NUM
ma-117	428	11	,	,	PUNCT
ma-117	428	12	milerr	milerr	NOUN
ma-117	428	13	=	=	SYM
ma-117	428	14	0.0022	0.0022	NUM
ma-117	428	15	and	and	CCONJ
ma-117	428	16	iemerr	iemerr	NOUN
ma-117	428	17	=	=	SYM
ma-117	428	18	0.0026	0.0026	NUM
ma-117	428	19	.	.	PUNCT
ma-117	428	20	remark	remark	PROPN
ma-117	428	21	5.1	5.1	NUM
ma-117	428	22	.	.	PUNCT
ma-117	429	1	from	from	ADP
ma-117	429	2	the	the	DET
ma-117	429	3	results	result	NOUN
ma-117	429	4	bellow	bellow	ADJ
ma-117	429	5	,	,	PUNCT
ma-117	429	6	in	in	ADP
ma-117	429	7	the	the	DET
ma-117	429	8	cases	case	NOUN
ma-117	429	9	of	of	ADP
ma-117	429	10	increasing	increase	VERB
ma-117	429	11	and	and	CCONJ
ma-117	429	12	decreasing	decrease	VERB
ma-117	429	13	stabilities	stability	NOUN
ma-117	429	14	of	of	ADP
ma-117	429	15	vasicek	vasicek	PROPN
ma-117	429	16	et	et	PROPN
ma-117	429	17	geometric	geometric	ADJ
ma-117	429	18	brownian	brownian	PROPN
ma-117	429	19	motion	motion	NOUN
ma-117	429	20	,	,	PUNCT
ma-117	429	21	we	we	PRON
ma-117	429	22	have	have	VERB
ma-117	429	23	that	that	PRON
ma-117	429	24	,	,	PUNCT
ma-117	429	25	the	the	DET
ma-117	429	26	milshtein	milshtein	PROPN
ma-117	429	27	scheme	scheme	NOUN
ma-117	429	28	is	be	AUX
ma-117	429	29	the	the	DET
ma-117	429	30	best	good	ADJ
ma-117	429	31	scheme	scheme	NOUN
ma-117	429	32	because	because	SCONJ
ma-117	429	33	it	it	PRON
ma-117	429	34	’s	’	VERB
ma-117	429	35	the	the	DET
ma-117	429	36	best	good	ADJ
ma-117	429	37	approximates	approximate	VERB
ma-117	429	38	the	the	DET
ma-117	429	39	exact	exact	ADJ
ma-117	429	40	solution	solution	NOUN
ma-117	429	41	.	.	PUNCT
ma-117	430	1	6	6	X
ma-117	430	2	.	.	X
ma-117	430	3	conclusion	conclusion	NOUN
ma-117	430	4	we	we	PRON
ma-117	430	5	have	have	AUX
ma-117	430	6	presented	present	VERB
ma-117	430	7	in	in	ADP
ma-117	430	8	this	this	DET
ma-117	430	9	article	article	NOUN
ma-117	430	10	the	the	DET
ma-117	430	11	analysis	analysis	NOUN
ma-117	430	12	of	of	ADP
ma-117	430	13	the	the	DET
ma-117	430	14	stability	stability	NOUN
ma-117	430	15	in	in	ADP
ma-117	430	16	mean	mean	ADJ
ma-117	430	17	and	and	CCONJ
ma-117	430	18	mean	mean	ADJ
ma-117	430	19	-	-	PUNCT
ma-117	430	20	square	square	ADJ
ma-117	430	21	forvasicek	forvasicek	NOUN
ma-117	430	22	and	and	CCONJ
ma-117	430	23	geometric	geometric	ADJ
ma-117	430	24	brownian	brownian	ADJ
ma-117	430	25	motion	motion	NOUN
ma-117	430	26	models	model	NOUN
ma-117	430	27	.	.	PUNCT
ma-117	431	1	in	in	ADP
ma-117	431	2	these	these	DET
ma-117	431	3	models	model	NOUN
ma-117	431	4	,	,	PUNCT
ma-117	431	5	we	we	PRON
ma-117	431	6	established	establish	VERB
ma-117	431	7	the	the	DET
ma-117	431	8	conditionsof	conditionsof	NOUN
ma-117	431	9	the	the	DET
ma-117	431	10	numerical	numerical	ADJ
ma-117	431	11	stabilities	stability	NOUN
ma-117	431	12	of	of	ADP
ma-117	431	13	euler	euler	NOUN
ma-117	431	14	-	-	PUNCT
ma-117	431	15	maruyama	maruyama	NOUN
ma-117	431	16	,	,	PUNCT
ma-117	431	17	implicit	implicit	ADJ
ma-117	431	18	euler	euler	NOUN
ma-117	431	19	-	-	PUNCT
ma-117	431	20	maruyama	maruyama	NOUN
ma-117	431	21	and	and	CCONJ
ma-117	431	22	milshtein	milshtein	NOUN
ma-117	431	23	schemes.these	schemes.these	PRON
ma-117	431	24	conditions	condition	NOUN
ma-117	431	25	have	have	AUX
ma-117	431	26	been	be	AUX
ma-117	431	27	proved	prove	VERB
ma-117	431	28	by	by	ADP
ma-117	431	29	using	use	VERB
ma-117	431	30	classical	classical	ADJ
ma-117	431	31	manner	manner	NOUN
ma-117	431	32	and	and	CCONJ
ma-117	431	33	y.	y.	PROPN
ma-117	431	34	saito	saito	PROPN
ma-117	431	35	’s	’s	PART
ma-117	431	36	approach	approach	NOUN
ma-117	431	37	.	.	PUNCT
ma-117	432	1	it	it	PRON
ma-117	432	2	should	should	AUX
ma-117	432	3	benoted	benote	VERB
ma-117	432	4	that	that	SCONJ
ma-117	432	5	each	each	DET
ma-117	432	6	case	case	NOUN
ma-117	432	7	is	be	AUX
ma-117	432	8	different	different	ADJ
ma-117	432	9	from	from	ADP
ma-117	432	10	the	the	DET
ma-117	432	11	other	other	ADJ
ma-117	432	12	depending	depend	VERB
ma-117	432	13	on	on	ADP
ma-117	432	14	whether	whether	SCONJ
ma-117	432	15	the	the	DET
ma-117	432	16	models	model	NOUN
ma-117	432	17	examined	examine	VERB
ma-117	432	18	have	have	AUX
ma-117	432	19	https://doi.org/10.28924/ada/ma.3.8	https://doi.org/10.28924/ada/ma.3.8	VERB
ma-117	432	20	eur	eur	ADJ
ma-117	432	21	.	.	PUNCT
ma-117	433	1	j.	j.	PROPN
ma-117	433	2	math	math	PROPN
ma-117	433	3	.	.	PUNCT
ma-117	434	1	anal	anal	PROPN
ma-117	434	2	.	.	PUNCT
ma-117	435	1	10.28924	10.28924	NUM
ma-117	435	2	/	/	SYM
ma-117	435	3	ada	ada	PROPN
ma-117	435	4	/	/	SYM
ma-117	435	5	ma.3.8	ma.3.8	PROPN
ma-117	435	6	21additive	21additive	NUM
ma-117	435	7	(	(	PUNCT
ma-117	435	8	vasicek	vasicek	PROPN
ma-117	435	9	model	model	NOUN
ma-117	435	10	)	)	PUNCT
ma-117	435	11	or	or	CCONJ
ma-117	435	12	multiplicative	multiplicative	ADJ
ma-117	435	13	(	(	PUNCT
ma-117	435	14	geometric	geometric	ADJ
ma-117	435	15	brownian	brownian	ADJ
ma-117	435	16	motion	motion	NOUN
ma-117	435	17	)	)	PUNCT
ma-117	435	18	white	white	ADJ
ma-117	435	19	noise	noise	NOUN
ma-117	435	20	type	type	NOUN
ma-117	435	21	.	.	PUNCT
ma-117	436	1	finally	finally	ADV
ma-117	436	2	,	,	PUNCT
ma-117	436	3	for	for	ADP
ma-117	436	4	these	these	DET
ma-117	436	5	models	model	NOUN
ma-117	436	6	,	,	PUNCT
ma-117	436	7	we	we	PRON
ma-117	436	8	found	find	VERB
ma-117	436	9	that	that	SCONJ
ma-117	436	10	the	the	DET
ma-117	436	11	stability	stability	NOUN
ma-117	436	12	conditions	condition	NOUN
ma-117	436	13	of	of	ADP
ma-117	436	14	the	the	DET
ma-117	436	15	vasicek	vasicek	PROPN
ma-117	436	16	model	model	NOUN
ma-117	436	17	coincideswith	coincideswith	ADP
ma-117	436	18	the	the	DET
ma-117	436	19	stability	stability	NOUN
ma-117	436	20	of	of	ADP
ma-117	436	21	the	the	DET
ma-117	436	22	odes	ode	NOUN
ma-117	436	23	,	,	PUNCT
ma-117	436	24	on	on	ADP
ma-117	436	25	the	the	DET
ma-117	436	26	other	other	ADJ
ma-117	436	27	hand	hand	NOUN
ma-117	436	28	,	,	PUNCT
ma-117	436	29	for	for	ADP
ma-117	436	30	the	the	DET
ma-117	436	31	stability	stability	NOUN
ma-117	436	32	conditions	condition	NOUN
ma-117	436	33	of	of	ADP
ma-117	436	34	the	the	DET
ma-117	436	35	second	second	ADJ
ma-117	436	36	modelto	modelto	NOUN
ma-117	436	37	coincide	coincide	VERB
ma-117	436	38	with	with	ADP
ma-117	436	39	the	the	DET
ma-117	436	40	stability	stability	NOUN
ma-117	436	41	of	of	ADP
ma-117	436	42	the	the	DET
ma-117	436	43	odes	ode	NOUN
ma-117	436	44	,	,	PUNCT
ma-117	436	45	it	it	PRON
ma-117	436	46	is	be	AUX
ma-117	436	47	necessary	necessary	ADJ
ma-117	436	48	that	that	SCONJ
ma-117	436	49	θ2	θ2	ADV
ma-117	436	50	<	<	X
ma-117	436	51	0	0	NUM
ma-117	436	52	.	.	PUNCT
ma-117	436	53	to	to	PART
ma-117	436	54	support	support	VERB
ma-117	436	55	these	these	DET
ma-117	436	56	results	result	NOUN
ma-117	436	57	,	,	PUNCT
ma-117	436	58	numerical	numerical	ADJ
ma-117	436	59	simulations	simulation	NOUN
ma-117	436	60	were	be	AUX
ma-117	436	61	made	make	VERB
ma-117	436	62	and	and	CCONJ
ma-117	436	63	the	the	DET
ma-117	436	64	calculations	calculation	NOUN
ma-117	436	65	of	of	ADP
ma-117	436	66	the	the	DET
ma-117	436	67	residuals	residual	NOUN
ma-117	436	68	(	(	PUNCT
ma-117	436	69	errors	error	NOUN
ma-117	436	70	)	)	PUNCT
ma-117	436	71	comes	come	VERB
ma-117	436	72	in	in	ADP
ma-117	436	73	support	support	NOUN
ma-117	436	74	ofthe	ofthe	NOUN
ma-117	436	75	results	result	NOUN
ma-117	436	76	found	find	VERB
ma-117	436	77	.	.	PUNCT
ma-117	437	1	in	in	ADP
ma-117	437	2	the	the	DET
ma-117	437	3	next	next	ADJ
ma-117	437	4	work	work	NOUN
ma-117	437	5	we	we	PRON
ma-117	437	6	will	will	AUX
ma-117	437	7	analyze	analyze	VERB
ma-117	437	8	the	the	DET
ma-117	437	9	numerical	numerical	ADJ
ma-117	437	10	stabilities	stability	NOUN
ma-117	437	11	of	of	ADP
ma-117	437	12	these	these	DET
ma-117	437	13	two	two	NUM
ma-117	437	14	modelsby	modelsby	ADJ
ma-117	437	15	using	use	VERB
ma-117	437	16	non	non	ADJ
ma-117	437	17	-	-	ADJ
ma-117	437	18	standard	standard	ADJ
ma-117	437	19	euler	euler	NOUN
ma-117	437	20	-	-	PUNCT
ma-117	437	21	maruyama	maruyama	NOUN
ma-117	437	22	scheme	scheme	NOUN
ma-117	437	23	.	.	PUNCT
ma-117	438	1	references	reference	NOUN
ma-117	438	2	[	[	X
ma-117	438	3	1	1	NUM
ma-117	438	4	]	]	PUNCT
ma-117	438	5	a.m.	a.m.	NOUN
ma-117	439	1	lyapunov	lyapunov	PROPN
ma-117	439	2	,	,	PUNCT
ma-117	439	3	the	the	DET
ma-117	439	4	general	general	ADJ
ma-117	439	5	problem	problem	NOUN
ma-117	439	6	of	of	ADP
ma-117	439	7	the	the	DET
ma-117	439	8	stability	stability	NOUN
ma-117	439	9	of	of	ADP
ma-117	439	10	motion	motion	NOUN
ma-117	439	11	,	,	PUNCT
ma-117	439	12	int	int	NOUN
ma-117	439	13	.	.	PUNCT
ma-117	440	1	j.	j.	PROPN
ma-117	440	2	control	control	PROPN
ma-117	440	3	.	.	PUNCT
ma-117	441	1	55	55	NUM
ma-117	441	2	(	(	PUNCT
ma-117	441	3	1992	1992	NUM
ma-117	441	4	)	)	PUNCT
ma-117	441	5	531?534	531?534	NOUN
ma-117	441	6	.	.	PUNCT
ma-117	442	1	https://doi	https://doi	NOUN
ma-117	442	2	.	.	PUNCT
ma-117	443	1	org/10.1080/00207179208934253.[2	org/10.1080/00207179208934253.[2	PROPN
ma-117	443	2	]	]	X
ma-117	443	3	i.	i.	PROPN
ma-117	443	4	kats	kats	PROPN
ma-117	443	5	,	,	PUNCT
ma-117	443	6	on	on	ADP
ma-117	443	7	the	the	DET
ma-117	443	8	stability	stability	NOUN
ma-117	443	9	in	in	ADP
ma-117	443	10	first	first	ADJ
ma-117	443	11	approximation	approximation	NOUN
ma-117	443	12	of	of	ADP
ma-117	443	13	systems	system	NOUN
ma-117	443	14	with	with	ADP
ma-117	443	15	random	random	ADJ
ma-117	443	16	lag	lag	NOUN
ma-117	443	17	,	,	PUNCT
ma-117	443	18	j.	j.	PROPN
ma-117	443	19	appl	appl	PROPN
ma-117	443	20	.	.	PROPN
ma-117	443	21	math	math	PROPN
ma-117	443	22	.	.	PUNCT
ma-117	444	1	mech	mech	PROPN
ma-117	444	2	.	.	PUNCT
ma-117	445	1	31	31	NUM
ma-117	445	2	(	(	PUNCT
ma-117	445	3	1967	1967	NUM
ma-117	445	4	)	)	PUNCT
ma-117	445	5	478?482	478?482	NUM
ma-117	445	6	.	.	PUNCT
ma-117	446	1	https://doi.org/10.1016/0021-8928(67)90030-5.[3	https://doi.org/10.1016/0021-8928(67)90030-5.[3	PROPN
ma-117	446	2	]	]	X
ma-117	446	3	i.i	i.i	PROPN
ma-117	446	4	.	.	PROPN
ma-117	446	5	gihman	gihman	PROPN
ma-117	446	6	,	,	PUNCT
ma-117	446	7	a.v	a.v	PROPN
ma-117	446	8	.	.	PROPN
ma-117	446	9	skorohod	skorohod	ADJ
ma-117	446	10	,	,	PUNCT
ma-117	446	11	stochastic	stochastic	ADJ
ma-117	446	12	differential	differential	ADJ
ma-117	446	13	equations	equation	NOUN
ma-117	446	14	,	,	PUNCT
ma-117	446	15	springer	springer	NOUN
ma-117	446	16	berlin	berlin	PROPN
ma-117	446	17	heidelberg	heidelberg	PROPN
ma-117	446	18	,	,	PUNCT
ma-117	446	19	1972	1972	NUM
ma-117	446	20	.	.	PUNCT
ma-117	447	1	https://doi.org/	https://doi.org/	VERB
ma-117	447	2	10.1007/978	10.1007/978	NUM
ma-117	447	3	-	-	SYM
ma-117	447	4	3	3	NUM
ma-117	447	5	-	-	PUNCT
ma-117	447	6	642	642	NUM
ma-117	447	7	-	-	PUNCT
ma-117	447	8	88264	88264	NUM
ma-117	447	9	-	-	SYM
ma-117	447	10	7.[4	7.[4	NOUN
ma-117	447	11	]	]	X
ma-117	447	12	a.	a.	NOUN
ma-117	447	13	friedman	friedman	PROPN
ma-117	447	14	,	,	PUNCT
ma-117	447	15	stochastic	stochastic	ADJ
ma-117	447	16	differential	differential	ADJ
ma-117	447	17	equations	equation	NOUN
ma-117	447	18	,	,	PUNCT
ma-117	447	19	in	in	ADP
ma-117	447	20	:	:	PUNCT
ma-117	447	21	stochastic	stochastic	ADJ
ma-117	447	22	differential	differential	ADJ
ma-117	447	23	equations	equation	NOUN
ma-117	447	24	and	and	CCONJ
ma-117	447	25	applications	application	NOUN
ma-117	447	26	,	,	PUNCT
ma-117	447	27	elsevier,1975	elsevier,1975	ADV
ma-117	447	28	:	:	PUNCT
ma-117	447	29	pp	pp	ADJ
ma-117	447	30	.	.	PUNCT
ma-117	448	1	98?127	98?127	NOUN
ma-117	448	2	.	.	PUNCT
ma-117	449	1	https://doi.org/10.1016/b978-0-12-268201-8.50010-4.[5	https://doi.org/10.1016/b978-0-12-268201-8.50010-4.[5	PUNCT
ma-117	449	2	]	]	X
ma-117	450	1	y.	y.	PROPN
ma-117	450	2	saito	saito	PROPN
ma-117	450	3	,	,	PUNCT
ma-117	450	4	stability	stability	NOUN
ma-117	450	5	analysis	analysis	NOUN
ma-117	450	6	of	of	ADP
ma-117	450	7	numerical	numerical	ADJ
ma-117	450	8	methods	method	NOUN
ma-117	450	9	for	for	ADP
ma-117	450	10	stochastic	stochastic	ADJ
ma-117	450	11	systems	system	NOUN
ma-117	450	12	with	with	ADP
ma-117	450	13	additive	additive	ADJ
ma-117	450	14	noise	noise	NOUN
ma-117	450	15	,	,	PUNCT
ma-117	450	16	rev	rev	PROPN
ma-117	450	17	.	.	PROPN
ma-117	450	18	econ	econ	PROPN
ma-117	450	19	.	.	PUNCT
ma-117	450	20	inf	inf	PROPN
ma-117	450	21	.	.	PUNCT
ma-117	450	22	stud	stud	PROPN
ma-117	450	23	.	.	PUNCT
ma-117	451	1	8(2008	8(2008	NOUN
ma-117	451	2	)	)	PUNCT
ma-117	451	3	119	119	NUM
ma-117	451	4	-	-	SYM
ma-117	451	5	123.[6	123.[6	NUM
ma-117	451	6	]	]	X
ma-117	451	7	j.p	j.p	PROPN
ma-117	451	8	.	.	PROPN
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ma-117	451	10	,	,	PUNCT
ma-117	451	11	mathematiques	mathematique	NOUN
ma-117	451	12	pour	pour	VERB
ma-117	451	13	les	les	PROPN
ma-117	451	14	systemes	systeme	NOUN
ma-117	451	15	dynamiques	dynamique	NOUN
ma-117	451	16	,	,	PUNCT
ma-117	451	17	hermes	hermes	NOUN
ma-117	451	18	science	science	NOUN
ma-117	451	19	,	,	PUNCT
ma-117	451	20	(	(	PUNCT
ma-117	451	21	2009).[7	2009).[7	X
ma-117	451	22	]	]	X
ma-117	451	23	g.n	g.n	PROPN
ma-117	451	24	.	.	PROPN
ma-117	451	25	milshtein	milshtein	PROPN
ma-117	451	26	,	,	PUNCT
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ma-117	451	28	motion	motion	NOUN
ma-117	451	29	,	,	PUNCT
ma-117	451	30	stochastic	stochastic	ADJ
ma-117	451	31	stability	stability	NOUN
ma-117	451	32	and	and	CCONJ
ma-117	451	33	differential	differential	ADJ
ma-117	451	34	equations	equation	NOUN
ma-117	451	35	,	,	PUNCT
ma-117	451	36	second	second	ADJ
ma-117	451	37	edition	edition	NOUN
ma-117	451	38	,	,	PUNCT
ma-117	451	39	moscow	moscow	PROPN
ma-117	451	40	3	3	NUM
ma-117	451	41	,	,	PUNCT
ma-117	451	42	(	(	PUNCT
ma-117	451	43	2011).[8	2011).[8	NOUN
ma-117	451	44	]	]	X
ma-117	451	45	t.c	t.c	PROPN
ma-117	451	46	.	.	PROPN
ma-117	451	47	gard	gard	PROPN
ma-117	451	48	,	,	PUNCT
ma-117	451	49	introduction	introduction	NOUN
ma-117	451	50	to	to	ADP
ma-117	451	51	stochastic	stochastic	ADJ
ma-117	451	52	differential	differential	ADJ
ma-117	451	53	equations	equation	NOUN
ma-117	451	54	,	,	PUNCT
ma-117	451	55	m.	m.	NOUN
ma-117	451	56	dekker	dekker	PROPN
ma-117	451	57	,	,	PUNCT
ma-117	451	58	new	new	PROPN
ma-117	451	59	york	york	PROPN
ma-117	451	60	,	,	PUNCT
ma-117	451	61	1988	1988	NUM
ma-117	451	62	.	.	PUNCT
ma-117	452	1	https://openlibrary	https://openlibrary	ADJ
ma-117	452	2	.	.	PUNCT
ma-117	452	3	org	org	ADJ
ma-117	452	4	/	/	SYM
ma-117	452	5	books	book	NOUN
ma-117	452	6	/	/	SYM
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ma-117	452	8	]	]	X
ma-117	452	9	e.	e.	PROPN
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ma-117	452	11	,	,	PUNCT
ma-117	452	12	n.	n.	PROPN
ma-117	452	13	ikeda	ikeda	PROPN
ma-117	452	14	,	,	PUNCT
ma-117	452	15	s.	s.	PROPN
ma-117	452	16	watanabe	watanabe	PROPN
ma-117	452	17	,	,	PUNCT
ma-117	452	18	stochastic	stochastic	ADJ
ma-117	452	19	differential	differential	ADJ
ma-117	452	20	equations	equation	NOUN
ma-117	452	21	and	and	CCONJ
ma-117	452	22	diffusion	diffusion	NOUN
ma-117	452	23	processes	process	NOUN
ma-117	452	24	,	,	PUNCT
ma-117	452	25	second	second	ADJ
ma-117	452	26	edition	edition	NOUN
ma-117	452	27	.	.	PUNCT
ma-117	453	1	north	north	PROPN
ma-117	453	2	-	-	PUNCT
ma-117	453	3	holland	holland	PROPN
ma-117	453	4	mathcmnticnl	mathcmnticnl	PROPN
ma-117	453	5	library	library	PROPN
ma-117	453	6	,	,	PUNCT
ma-117	453	7	amsterdam	amsterdam	PROPN
ma-117	453	8	,	,	PUNCT
ma-117	453	9	(	(	PUNCT
ma-117	453	10	1989).[10	1989).[10	NUM
ma-117	453	11	]	]	X
ma-117	453	12	p.e	p.e	PROPN
ma-117	453	13	.	.	PROPN
ma-117	453	14	kloden	kloden	PROPN
ma-117	453	15	,	,	PUNCT
ma-117	453	16	e.	e.	PROPN
ma-117	453	17	platen	platen	PROPN
ma-117	453	18	,	,	PUNCT
ma-117	453	19	numerical	numerical	ADJ
ma-117	453	20	solution	solution	NOUN
ma-117	453	21	of	of	ADP
ma-117	453	22	stochastic	stochastic	ADJ
ma-117	453	23	differential	differential	ADJ
ma-117	453	24	equations	equation	NOUN
ma-117	453	25	,	,	PUNCT
ma-117	453	26	springer	springer	NOUN
ma-117	453	27	-	-	PUNCT
ma-117	453	28	verlag	verlag	PROPN
ma-117	453	29	berlin	berlin	PROPN
ma-117	453	30	heidelbeg,(2003).[11	heidelbeg,(2003).[11	PROPN
ma-117	453	31	]	]	PUNCT
ma-117	453	32	p.e	p.e	PROPN
ma-117	453	33	.	.	PROPN
ma-117	453	34	kloeden	kloeden	PROPN
ma-117	453	35	,	,	PUNCT
ma-117	453	36	e.	e.	PROPN
ma-117	453	37	platen	platen	PROPN
ma-117	453	38	,	,	PUNCT
ma-117	453	39	numerical	numerical	ADJ
ma-117	453	40	solution	solution	NOUN
ma-117	453	41	of	of	ADP
ma-117	453	42	stochastic	stochastic	ADJ
ma-117	453	43	differential	differential	ADJ
ma-117	453	44	equations	equation	NOUN
ma-117	453	45	,	,	PUNCT
ma-117	453	46	springer	springer	NOUN
ma-117	453	47	berlin	berlin	PROPN
ma-117	453	48	heidelberg	heidelberg	PROPN
ma-117	453	49	,	,	PUNCT
ma-117	453	50	(	(	PUNCT
ma-117	453	51	2011).[12	2011).[12	NUM
ma-117	453	52	]	]	X
ma-117	453	53	x.	x.	NOUN
ma-117	453	54	mao	mao	PROPN
ma-117	453	55	,	,	PUNCT
ma-117	453	56	stochastic	stochastic	ADJ
ma-117	453	57	differential	differential	ADJ
ma-117	453	58	equations	equation	NOUN
ma-117	453	59	and	and	CCONJ
ma-117	453	60	application	application	NOUN
ma-117	453	61	,	,	PUNCT
ma-117	453	62	horwood	horwood	NOUN
ma-117	453	63	,	,	PUNCT
ma-117	453	64	chichester	chichester	PROPN
ma-117	453	65	,	,	PUNCT
ma-117	453	66	(	(	PUNCT
ma-117	453	67	1997).[13	1997).[13	PROPN
ma-117	453	68	]	]	X
ma-117	453	69	b.	b.	PROPN
ma-117	453	70	oksendal	oksendal	PROPN
ma-117	453	71	,	,	PUNCT
ma-117	453	72	stochastic	stochastic	ADJ
ma-117	453	73	differential	differential	ADJ
ma-117	453	74	equations	equation	NOUN
ma-117	453	75	,	,	PUNCT
ma-117	453	76	sixth	sixth	ADJ
ma-117	453	77	edition	edition	NOUN
ma-117	453	78	,	,	PUNCT
ma-117	453	79	springer	springer	NOUN
ma-117	453	80	,	,	PUNCT
ma-117	453	81	berlin	berlin	PROPN
ma-117	453	82	,	,	PUNCT
ma-117	453	83	heidelberg	heidelberg	PROPN
ma-117	453	84	,	,	PUNCT
ma-117	453	85	(	(	PUNCT
ma-117	453	86	2003).[14	2003).[14	PROPN
ma-117	453	87	]	]	X
ma-117	453	88	b.	b.	PROPN
ma-117	453	89	oksendal	oksendal	PROPN
ma-117	453	90	,	,	PUNCT
ma-117	453	91	s.	s.	PROPN
ma-117	453	92	agnes	agnes	PROPN
ma-117	453	93	,	,	PUNCT
ma-117	453	94	applied	apply	VERB
ma-117	453	95	stochastic	stochastic	ADJ
ma-117	453	96	control	control	NOUN
ma-117	453	97	of	of	ADP
ma-117	453	98	jump	jump	NOUN
ma-117	453	99	diffusions	diffusion	NOUN
ma-117	453	100	,	,	PUNCT
ma-117	453	101	springer	springer	NOUN
ma-117	453	102	berlin	berlin	PROPN
ma-117	453	103	heidelberg	heidelberg	PROPN
ma-117	453	104	,	,	PUNCT
ma-117	453	105	(	(	PUNCT
ma-117	453	106	2007	2007	NUM
ma-117	453	107	)	)	PUNCT
ma-117	453	108	.	.	PUNCT
ma-117	454	1	https	https	NOUN
ma-117	454	2	:	:	PUNCT
ma-117	454	3	//doi.org/10.1007/978	//doi.org/10.1007/978	NUM
ma-117	454	4	-	-	PUNCT
ma-117	454	5	3	3	NUM
ma-117	454	6	-	-	PUNCT
ma-117	454	7	540	540	NUM
ma-117	454	8	-	-	PUNCT
ma-117	454	9	69826	69826	NUM
ma-117	454	10	-	-	SYM
ma-117	454	11	5.[15	5.[15	NUM
ma-117	454	12	]	]	X
ma-117	454	13	s.m	s.m	PROPN
ma-117	454	14	.	.	PROPN
ma-117	454	15	iacus	iacus	PROPN
ma-117	454	16	,	,	PUNCT
ma-117	454	17	simulation	simulation	NOUN
ma-117	454	18	and	and	CCONJ
ma-117	454	19	inference	inference	NOUN
ma-117	454	20	for	for	ADP
ma-117	454	21	stochastic	stochastic	ADJ
ma-117	454	22	differential	differential	ADJ
ma-117	454	23	equations	equation	NOUN
ma-117	454	24	,	,	PUNCT
ma-117	454	25	springer	springer	NOUN
ma-117	454	26	new	new	PROPN
ma-117	454	27	york	york	PROPN
ma-117	454	28	,	,	PUNCT
ma-117	454	29	(	(	PUNCT
ma-117	454	30	2008	2008	NUM
ma-117	454	31	)	)	PUNCT
ma-117	454	32	.	.	PUNCT
ma-117	455	1	https	https	NOUN
ma-117	455	2	:	:	PUNCT
ma-117	455	3	//doi.org/10.1007/978	//doi.org/10.1007/978	NUM
ma-117	455	4	-	-	SYM
ma-117	455	5	0	0	NUM
ma-117	455	6	-	-	PUNCT
ma-117	455	7	387	387	NUM
ma-117	455	8	-	-	NUM
ma-117	455	9	75839	75839	NUM
ma-117	455	10	-	-	SYM
ma-117	455	11	8.[16	8.[16	NUM
ma-117	455	12	]	]	X
ma-117	455	13	d.	d.	PROPN
ma-117	455	14	sondermann	sondermann	PROPN
ma-117	455	15	,	,	PUNCT
ma-117	455	16	introduction	introduction	NOUN
ma-117	455	17	to	to	ADP
ma-117	455	18	stochastic	stochastic	ADJ
ma-117	455	19	calculus	calculus	NOUN
ma-117	455	20	for	for	ADP
ma-117	455	21	finance	finance	NOUN
ma-117	455	22	,	,	PUNCT
ma-117	455	23	springer	springer	NOUN
ma-117	455	24	berlin	berlin	PROPN
ma-117	455	25	heidelberg	heidelberg	PROPN
ma-117	455	26	,	,	PUNCT
ma-117	455	27	(	(	PUNCT
ma-117	455	28	2006	2006	NUM
ma-117	455	29	)	)	PUNCT
ma-117	455	30	.	.	PUNCT
ma-117	456	1	https://doi	https://doi	X
ma-117	456	2	.	.	PUNCT
ma-117	457	1	org/10.1007/3	org/10.1007/3	NUM
ma-117	457	2	-	-	PUNCT
ma-117	457	3	540	540	NUM
ma-117	457	4	-	-	PUNCT
ma-117	457	5	34837	34837	NUM
ma-117	457	6	-	-	SYM
ma-117	457	7	9.[17	9.[17	NUM
ma-117	457	8	]	]	X
ma-117	457	9	y.	y.	PROPN
ma-117	457	10	komori	komori	PROPN
ma-117	457	11	,	,	PUNCT
ma-117	457	12	stahle	stahle	NOUN
ma-117	457	13	row	row	NOUN
ma-117	457	14	-	-	PUNCT
ma-117	457	15	type	type	NOUN
ma-117	457	16	weak	weak	ADJ
ma-117	457	17	scheme	scheme	NOUN
ma-117	457	18	for	for	ADP
ma-117	457	19	stochastic	stochastic	ADJ
ma-117	457	20	differential	differential	ADJ
ma-117	457	21	equations	equation	NOUN
ma-117	457	22	,	,	PUNCT
ma-117	457	23	monte	monte	PROPN
ma-117	457	24	carlo	carlo	PROPN
ma-117	457	25	methods	method	NOUN
ma-117	457	26	appl	appl	PROPN
ma-117	457	27	.	.	PUNCT
ma-117	458	1	1(1995	1(1995	NUM
ma-117	458	2	)	)	PUNCT
ma-117	458	3	279	279	NUM
ma-117	458	4	-	-	SYM
ma-117	458	5	300	300	NUM
ma-117	458	6	.	.	PUNCT
ma-117	459	1	https://doi.org/10.1515/mcma.1995.1.4.279.[18	https://doi.org/10.1515/mcma.1995.1.4.279.[18	NOUN
ma-117	459	2	]	]	X
ma-117	459	3	y.	y.	PROPN
ma-117	459	4	komori	komori	PROPN
ma-117	459	5	,	,	PUNCT
ma-117	459	6	y.	y.	PROPN
ma-117	459	7	saito	saito	PROPN
ma-117	459	8	,	,	PUNCT
ma-117	459	9	t.	t.	PROPN
ma-117	459	10	mitsui	mitsui	PROPN
ma-117	459	11	,	,	PUNCT
ma-117	459	12	some	some	DET
ma-117	459	13	issues	issue	NOUN
ma-117	459	14	in	in	ADP
ma-117	459	15	discrete	discrete	ADJ
ma-117	459	16	approximate	approximate	ADJ
ma-117	459	17	solution	solution	NOUN
ma-117	459	18	for	for	ADP
ma-117	459	19	stochastic	stochastic	ADJ
ma-117	459	20	differential	differential	ADJ
ma-117	459	21	equations	equation	NOUN
ma-117	459	22	,	,	PUNCT
ma-117	459	23	computers	computer	NOUN
ma-117	459	24	math	math	NOUN
ma-117	459	25	.	.	PUNCT
ma-117	460	1	appl	appl	PROPN
ma-117	460	2	.	.	PUNCT
ma-117	461	1	28	28	NUM
ma-117	461	2	(	(	PUNCT
ma-117	461	3	1994	1994	NUM
ma-117	461	4	)	)	PUNCT
ma-117	461	5	269?278	269?278	NUM
ma-117	461	6	.	.	PUNCT
ma-117	462	1	https://doi.org/10.1016/0898-1221(94)00197-9	https://doi.org/10.1016/0898-1221(94)00197-9	NOUN
ma-117	462	2	.	.	PUNCT
ma-117	463	1	https://doi.org/10.28924/ada/ma.3.8	https://doi.org/10.28924/ada/ma.3.8	PROPN
ma-117	463	2	https://doi.org/10.1080/00207179208934253	https://doi.org/10.1080/00207179208934253	PROPN
ma-117	463	3	https://doi.org/10.1080/00207179208934253	https://doi.org/10.1080/00207179208934253	PROPN
ma-117	463	4	https://doi.org/10.1016/0021-8928(67)90030-5	https://doi.org/10.1016/0021-8928(67)90030-5	PROPN
ma-117	463	5	https://doi.org/10.1007/978-3-642-88264-7	https://doi.org/10.1007/978-3-642-88264-7	SYM
ma-117	463	6	https://doi.org/10.1007/978-3-642-88264-7	https://doi.org/10.1007/978-3-642-88264-7	PROPN
ma-117	463	7	https://doi.org/10.1016/b978-0-12-268201-8.50010-4	https://doi.org/10.1016/b978-0-12-268201-8.50010-4	ADJ
ma-117	463	8	https://openlibrary.org/books/ol2391851	https://openlibrary.org/books/ol2391851	PROPN
ma-117	463	9	m	m	PROPN
ma-117	463	10	https://openlibrary.org/books/ol2391851	https://openlibrary.org/books/ol2391851	PROPN
ma-117	463	11	m	m	PROPN
ma-117	463	12	https://doi.org/10.1007/978-3-540-69826-5	https://doi.org/10.1007/978-3-540-69826-5	PROPN
ma-117	463	13	https://doi.org/10.1007/978-3-540-69826-5	https://doi.org/10.1007/978-3-540-69826-5	PROPN
ma-117	463	14	https://doi.org/10.1007/978-0-387-75839-8	https://doi.org/10.1007/978-0-387-75839-8	PROPN
ma-117	463	15	https://doi.org/10.1007/978-0-387-75839-8	https://doi.org/10.1007/978-0-387-75839-8	PROPN
ma-117	463	16	https://doi.org/10.1007/3-540-34837-9	https://doi.org/10.1007/3-540-34837-9	NOUN
ma-117	463	17	https://doi.org/10.1007/3-540-34837-9	https://doi.org/10.1007/3-540-34837-9	NOUN
ma-117	463	18	https://doi.org/10.1515/mcma.1995.1.4.279	https://doi.org/10.1515/mcma.1995.1.4.279	PROPN
ma-117	463	19	https://doi.org/10.1016/0898-1221(94)00197-9	https://doi.org/10.1016/0898-1221(94)00197-9	PROPN
ma-117	463	20	eur	eur	PROPN
ma-117	463	21	.	.	PUNCT
ma-117	464	1	j.	j.	PROPN
ma-117	464	2	math	math	PROPN
ma-117	464	3	.	.	PUNCT
ma-117	465	1	anal	anal	PROPN
ma-117	465	2	.	.	PUNCT
ma-117	466	1	10.28924	10.28924	NUM
ma-117	466	2	/	/	SYM
ma-117	466	3	ada	ada	PROPN
ma-117	466	4	/	/	SYM
ma-117	466	5	ma.3.8	ma.3.8	PROPN
ma-117	466	6	22	22	NUM
ma-117	467	1	[	[	X
ma-117	467	2	19	19	NUM
ma-117	467	3	]	]	X
ma-117	467	4	y.	y.	PROPN
ma-117	467	5	saito	saito	PROPN
ma-117	467	6	,	,	PUNCT
ma-117	467	7	t.	t.	PROPN
ma-117	467	8	mitsui	mitsui	PROPN
ma-117	467	9	,	,	PUNCT
ma-117	467	10	stability	stability	NOUN
ma-117	467	11	analysis	analysis	NOUN
ma-117	467	12	of	of	ADP
ma-117	467	13	numerical	numerical	ADJ
ma-117	467	14	schemes	scheme	NOUN
ma-117	467	15	for	for	ADP
ma-117	467	16	stochastic	stochastic	ADJ
ma-117	467	17	differential	differential	ADJ
ma-117	467	18	equations	equation	NOUN
ma-117	467	19	,	,	PUNCT
ma-117	467	20	siam	siam	PROPN
ma-117	467	21	j.	j.	PROPN
ma-117	467	22	numer.anal	numer.anal	PROPN
ma-117	467	23	.	.	PROPN
ma-117	468	1	33	33	NUM
ma-117	468	2	(	(	PUNCT
ma-117	468	3	1996	1996	NUM
ma-117	468	4	)	)	PUNCT
ma-117	468	5	2254	2254	NUM
ma-117	468	6	-	-	SYM
ma-117	468	7	2267	2267	NUM
ma-117	468	8	.	.	PUNCT
ma-117	469	1	https://doi.org/10.1137/s0036142992228409.[20	https://doi.org/10.1137/s0036142992228409.[20	X
ma-117	469	2	]	]	X
ma-117	469	3	k.	k.	PROPN
ma-117	469	4	burrage	burrage	PROPN
ma-117	469	5	,	,	PUNCT
ma-117	469	6	p.	p.	PROPN
ma-117	469	7	burrage	burrage	PROPN
ma-117	469	8	,	,	PUNCT
ma-117	469	9	t.	t.	PROPN
ma-117	469	10	mitsui	mitsui	PROPN
ma-117	469	11	,	,	PUNCT
ma-117	469	12	numerical	numerical	ADJ
ma-117	469	13	solutions	solution	NOUN
ma-117	469	14	of	of	ADP
ma-117	469	15	stochastic	stochastic	ADJ
ma-117	469	16	differential	differential	ADJ
ma-117	469	17	equations	equation	NOUN
ma-117	469	18	?	?	PUNCT
ma-117	470	1	implementation	implementation	NOUN
ma-117	470	2	and	and	CCONJ
ma-117	470	3	sta	sta	ADJ
ma-117	470	4	-	-	PUNCT
ma-117	470	5	bility	bility	NOUN
ma-117	470	6	issues	issue	NOUN
ma-117	470	7	,	,	PUNCT
ma-117	470	8	j.	j.	PROPN
ma-117	470	9	comput	comput	PROPN
ma-117	470	10	.	.	PUNCT
ma-117	471	1	appl	appl	PROPN
ma-117	471	2	.	.	PROPN
ma-117	471	3	math	math	NOUN
ma-117	471	4	.	.	PUNCT
ma-117	472	1	125	125	NUM
ma-117	472	2	(	(	PUNCT
ma-117	472	3	2000	2000	NUM
ma-117	472	4	)	)	PUNCT
ma-117	472	5	171?182	171?182	NOUN
ma-117	472	6	.	.	PUNCT
ma-117	473	1	https://doi.org/10.1016/s0377-0427(00)00467-2.[21	https://doi.org/10.1016/s0377-0427(00)00467-2.[21	PROPN
ma-117	473	2	]	]	X
ma-117	473	3	y.	y.	PROPN
ma-117	473	4	saito	saito	PROPN
ma-117	473	5	,	,	PUNCT
ma-117	473	6	t.	t.	PROPN
ma-117	473	7	mitsui	mitsui	PROPN
ma-117	473	8	,	,	PUNCT
ma-117	473	9	t	t	PROPN
ma-117	473	10	-	-	PUNCT
ma-117	473	11	stability	stability	NOUN
ma-117	473	12	of	of	ADP
ma-117	473	13	numerial	numerial	ADJ
ma-117	473	14	scheme	scheme	NOUN
ma-117	473	15	for	for	ADP
ma-117	473	16	stochastic	stochastic	ADJ
ma-117	473	17	differential	differential	ADJ
ma-117	473	18	equations	equation	NOUN
ma-117	473	19	,	,	PUNCT
ma-117	473	20	contribut	contribut	NOUN
ma-117	473	21	.	.	PUNCT
ma-117	474	1	numer	numer	PROPN
ma-117	474	2	.	.	PUNCT
ma-117	475	1	math.(1993	math.(1993	NOUN
ma-117	475	2	)	)	PUNCT
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ma-117	476	2	.	.	PUNCT
ma-117	477	1	https://doi.org/10.1142/9789812798886_0026.[22	https://doi.org/10.1142/9789812798886_0026.[22	NOUN
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ma-117	477	3	y.	y.	PROPN
ma-117	477	4	saito	saito	PROPN
ma-117	477	5	,	,	PUNCT
ma-117	477	6	t.	t.	PROPN
ma-117	477	7	mitsui	mitsui	PROPN
ma-117	477	8	,	,	PUNCT
ma-117	477	9	mean	mean	ADJ
ma-117	477	10	-	-	PUNCT
ma-117	477	11	square	square	ADJ
ma-117	477	12	stability	stability	NOUN
ma-117	477	13	of	of	ADP
ma-117	477	14	numerical	numerical	ADJ
ma-117	477	15	schemes	scheme	NOUN
ma-117	477	16	for	for	ADP
ma-117	477	17	stochastic	stochastic	ADJ
ma-117	477	18	differential	differential	ADJ
ma-117	477	19	systems	system	NOUN
ma-117	477	20	,	,	PUNCT
ma-117	477	21	vietnam	vietnam	PROPN
ma-117	477	22	j.	j.	PROPN
ma-117	477	23	math.30	math.30	PROPN
ma-117	477	24	(	(	PUNCT
ma-117	477	25	2002	2002	NUM
ma-117	477	26	)	)	PUNCT
ma-117	477	27	551	551	NUM
ma-117	477	28	-	-	SYM
ma-117	477	29	560.[23	560.[23	NUM
ma-117	477	30	]	]	X
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ma-117	477	32	sakthivel	sakthivel	PROPN
ma-117	477	33	,	,	PUNCT
ma-117	477	34	p.	p.	PROPN
ma-117	477	35	revathi	revathi	PROPN
ma-117	477	36	,	,	PUNCT
ma-117	477	37	n.i	n.i	PROPN
ma-117	477	38	.	.	PROPN
ma-117	477	39	mahmudov	mahmudov	PROPN
ma-117	477	40	,	,	PUNCT
ma-117	477	41	asymptotic	asymptotic	ADJ
ma-117	477	42	stability	stability	NOUN
ma-117	477	43	of	of	ADP
ma-117	477	44	fractional	fractional	ADJ
ma-117	477	45	stochastic	stochastic	ADJ
ma-117	477	46	neutral	neutral	ADJ
ma-117	477	47	differential	differential	NOUN
ma-117	477	48	equationswith	equationswith	PROPN
ma-117	477	49	infinite	infinite	ADJ
ma-117	477	50	delays	delay	NOUN
ma-117	477	51	,	,	PUNCT
ma-117	477	52	abstr	abstr	PROPN
ma-117	477	53	.	.	PUNCT
ma-117	477	54	appl	appl	PROPN
ma-117	477	55	.	.	PUNCT
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ma-117	478	2	.	.	PUNCT
ma-117	479	1	2013	2013	NUM
ma-117	479	2	(	(	PUNCT
ma-117	479	3	2013	2013	NUM
ma-117	479	4	)	)	PUNCT
ma-117	479	5	769257	769257	NUM
ma-117	479	6	.	.	PUNCT
ma-117	480	1	https://doi.org/10.1155/2013/769257.[24	https://doi.org/10.1155/2013/769257.[24	NOUN
ma-117	480	2	]	]	PUNCT
ma-117	480	3	x.	x.	NOUN
ma-117	480	4	mao	mao	PROPN
ma-117	480	5	,	,	PUNCT
ma-117	480	6	exponential	exponential	ADJ
ma-117	480	7	stability	stability	NOUN
ma-117	480	8	of	of	ADP
ma-117	480	9	large	large	ADJ
ma-117	480	10	-	-	PUNCT
ma-117	480	11	scale	scale	NOUN
ma-117	480	12	stochastic	stochastic	ADJ
ma-117	480	13	differential	differential	ADJ
ma-117	480	14	equations	equation	NOUN
ma-117	480	15	,	,	PUNCT
ma-117	480	16	syst	syst	PROPN
ma-117	480	17	.	.	PUNCT
ma-117	481	1	control	control	PROPN
ma-117	481	2	lett	lett	PROPN
ma-117	481	3	.	.	PROPN
ma-117	482	1	19	19	NUM
ma-117	482	2	(	(	PUNCT
ma-117	482	3	1992	1992	NUM
ma-117	482	4	)	)	PUNCT
ma-117	483	1	71?81	71?81	NOUN
ma-117	483	2	.	.	PUNCT
ma-117	484	1	https://doi.org/10.1016/0167-6911(92)90042-q.[25	https://doi.org/10.1016/0167-6911(92)90042-q.[25	PROPN
ma-117	484	2	]	]	X
ma-117	484	3	r.	r.	PROPN
ma-117	484	4	khasminskii	khasminskii	PROPN
ma-117	484	5	,	,	PUNCT
ma-117	484	6	stochastic	stochastic	ADJ
ma-117	484	7	stability	stability	NOUN
ma-117	484	8	of	of	ADP
ma-117	484	9	differential	differential	ADJ
ma-117	484	10	equations	equation	NOUN
ma-117	484	11	,	,	PUNCT
ma-117	484	12	springer	springer	NOUN
ma-117	484	13	berlin	berlin	PROPN
ma-117	484	14	heidelberg	heidelberg	PROPN
ma-117	484	15	,	,	PUNCT
ma-117	484	16	(	(	PUNCT
ma-117	484	17	2012	2012	NUM
ma-117	484	18	)	)	PUNCT
ma-117	484	19	.	.	PUNCT
ma-117	485	1	https://doi	https://doi	PROPN
ma-117	485	2	.	.	PUNCT
ma-117	485	3	org/10.1007/978	org/10.1007/978	PROPN
ma-117	485	4	-	-	PUNCT
ma-117	485	5	3	3	NUM
ma-117	485	6	-	-	PUNCT
ma-117	485	7	642	642	NUM
ma-117	485	8	-	-	PUNCT
ma-117	485	9	23280	23280	NUM
ma-117	485	10	-	-	PUNCT
ma-117	485	11	0.[26	0.[26	NUM
ma-117	485	12	]	]	X
ma-117	485	13	r.m	r.m	PROPN
ma-117	485	14	.	.	PROPN
ma-117	485	15	sheldon	sheldon	PROPN
ma-117	485	16	,	,	PUNCT
ma-117	485	17	variations	variation	NOUN
ma-117	485	18	sur	sur	PROPN
ma-117	485	19	le	le	X
ma-117	485	20	mouvement	mouvement	PROPN
ma-117	485	21	brownien	brownien	PROPN
ma-117	485	22	:	:	PUNCT
ma-117	485	23	introduction	introduction	NOUN
ma-117	485	24	aux	aux	PROPN
ma-117	485	25	modeles	modeles	PROPN
ma-117	485	26	de	de	X
ma-117	485	27	probabilite	probabilite	ADJ
ma-117	485	28	(	(	PUNCT
ma-117	485	29	11e	11e	NOUN
ma-117	485	30	edition	edition	NOUN
ma-117	485	31	)	)	PUNCT
ma-117	485	32	,	,	PUNCT
ma-117	485	33	elsevier	elsevier	NOUN
ma-117	485	34	,	,	PUNCT
ma-117	485	35	amsterdam	amsterdam	PROPN
ma-117	485	36	,	,	PUNCT
ma-117	485	37	(	(	PUNCT
ma-117	485	38	2014	2014	NUM
ma-117	485	39	)	)	PUNCT
ma-117	485	40	.	.	PUNCT
ma-117	486	1	https://doi.org/10.28924/ada/ma.3.8	https://doi.org/10.28924/ada/ma.3.8	AUX
ma-117	486	2	https://doi.org/10.1137/s0036142992228409	https://doi.org/10.1137/s0036142992228409	PRON
ma-117	486	3	https://doi.org/10.1016/s0377-0427(00)00467-2	https://doi.org/10.1016/s0377-0427(00)00467-2	VERB
ma-117	486	4	https://doi.org/10.1142/9789812798886_0026	https://doi.org/10.1142/9789812798886_0026	PROPN
ma-117	486	5	https://doi.org/10.1155/2013/769257	https://doi.org/10.1155/2013/769257	NOUN
ma-117	486	6	https://doi.org/10.1016/0167-6911(92)90042-q	https://doi.org/10.1016/0167-6911(92)90042-q	VERB
ma-117	486	7	https://doi.org/10.1007/978-3-642-23280-0	https://doi.org/10.1007/978-3-642-23280-0	PROPN
ma-117	486	8	https://doi.org/10.1007/978-3-642-23280-0	https://doi.org/10.1007/978-3-642-23280-0	PROPN
ma-117	486	9	1	1	NUM
ma-117	486	10	.	.	PUNCT
ma-117	487	1	introduction	introduction	NOUN
ma-117	487	2	2	2	NUM
ma-117	487	3	.	.	PUNCT
ma-117	487	4	preliminary	preliminary	ADJ
ma-117	487	5	notions	notion	NOUN
ma-117	487	6	2.1	2.1	NUM
ma-117	487	7	.	.	PUNCT
ma-117	488	1	stochastic	stochastic	ADJ
ma-117	488	2	differential	differential	ADJ
ma-117	488	3	equation	equation	NOUN
ma-117	488	4	and	and	CCONJ
ma-117	488	5	stabilities	stability	NOUN
ma-117	488	6	2.2	2.2	NUM
ma-117	488	7	.	.	PUNCT
ma-117	489	1	stochastic	stochastic	ADJ
ma-117	489	2	numerical	numerical	ADJ
ma-117	489	3	schemes	scheme	NOUN
ma-117	489	4	3	3	NUM
ma-117	489	5	.	.	PUNCT
ma-117	489	6	numerical	numerical	ADJ
ma-117	489	7	stabilities	stability	NOUN
ma-117	489	8	of	of	ADP
ma-117	489	9	vasicek	vasicek	PROPN
ma-117	489	10	model	model	NOUN
ma-117	489	11	3.1	3.1	NUM
ma-117	489	12	.	.	PUNCT
ma-117	489	13	explicit	explicit	ADJ
ma-117	489	14	solution	solution	NOUN
ma-117	489	15	3.2	3.2	NUM
ma-117	489	16	.	.	PUNCT
ma-117	489	17	euler	euler	NOUN
ma-117	489	18	-	-	PUNCT
ma-117	489	19	maruyama	maruyama	NOUN
ma-117	489	20	scheme	scheme	NOUN
ma-117	489	21	stabilities	stability	NOUN
ma-117	489	22	3.3	3.3	NUM
ma-117	489	23	.	.	PUNCT
ma-117	490	1	milshtein	milshtein	PROPN
ma-117	490	2	's	's	PART
ma-117	490	3	scheme	scheme	NOUN
ma-117	490	4	stabilities	stability	NOUN
ma-117	490	5	3.4	3.4	NUM
ma-117	490	6	.	.	PUNCT
ma-117	490	7	implicit	implicit	ADJ
ma-117	490	8	euler	euler	VERB
ma-117	490	9	-	-	PUNCT
ma-117	490	10	maruyama	maruyama	NOUN
ma-117	490	11	scheme	scheme	NOUN
ma-117	490	12	stabilities	stability	NOUN
ma-117	490	13	4	4	NUM
ma-117	490	14	.	.	PUNCT
ma-117	490	15	numerical	numerical	ADJ
ma-117	490	16	stabilities	stability	NOUN
ma-117	490	17	of	of	ADP
ma-117	490	18	geometric	geometric	ADJ
ma-117	490	19	brownian	brownian	ADJ
ma-117	490	20	motion	motion	NOUN
ma-117	490	21	4.1	4.1	NUM
ma-117	490	22	.	.	PUNCT
ma-117	490	23	explicit	explicit	ADJ
ma-117	490	24	solution	solution	NOUN
ma-117	490	25	of	of	ADP
ma-117	490	26	the	the	DET
ma-117	490	27	model	model	NOUN
ma-117	490	28	4.2	4.2	NUM
ma-117	490	29	.	.	PUNCT
ma-117	490	30	euler	euler	NOUN
ma-117	490	31	-	-	PUNCT
ma-117	490	32	maruyama	maruyama	NOUN
ma-117	490	33	scheme	scheme	NOUN
ma-117	490	34	stabilities	stability	NOUN
ma-117	490	35	4.3	4.3	NUM
ma-117	490	36	.	.	PUNCT
ma-117	491	1	milshtein	milshtein	PROPN
ma-117	491	2	's	's	PART
ma-117	491	3	scheme	scheme	NOUN
ma-117	491	4	stabilities	stability	NOUN
ma-117	491	5	4.4	4.4	NUM
ma-117	491	6	.	.	PUNCT
ma-117	492	1	implicit	implicit	ADJ
ma-117	492	2	euler	euler	VERB
ma-117	492	3	-	-	PUNCT
ma-117	492	4	maruyama	maruyama	NOUN
ma-117	492	5	scheme	scheme	NOUN
ma-117	492	6	stabilities	stability	NOUN
ma-117	492	7	5	5	NUM
ma-117	492	8	.	.	PUNCT
ma-117	492	9	numerical	numerical	ADJ
ma-117	492	10	simulations	simulation	NOUN
ma-117	492	11	and	and	CCONJ
ma-117	492	12	residual	residual	ADJ
ma-117	492	13	calculations	calculation	NOUN
ma-117	492	14	5.1	5.1	NUM
ma-117	492	15	.	.	PUNCT
ma-117	493	1	numerical	numerical	PROPN
ma-117	493	2	simulation	simulation	PROPN
ma-117	493	3	of	of	ADP
ma-117	493	4	vasicek	vasicek	PROPN
ma-117	493	5	and	and	CCONJ
ma-117	493	6	geometric	geometric	ADJ
ma-117	493	7	brownian	brownian	ADJ
ma-117	493	8	motion	motion	NOUN
ma-117	493	9	models	model	NOUN
ma-117	493	10	5.2	5.2	NUM
ma-117	493	11	.	.	PUNCT
ma-117	494	1	interpretation	interpretation	NOUN
ma-117	494	2	of	of	ADP
ma-117	494	3	results	result	NOUN
ma-117	494	4	6	6	NUM
ma-117	494	5	.	.	PUNCT
ma-117	495	1	conclusion	conclusion	NOUN
ma-117	495	2	references	reference	NOUN
ma-117	495	3	bibliographie	bibliographie	NOUN