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Numerical Stabilities of Vasicek and Geometric Brownian Motion Models

O. C. Badibi1,∗, I. Ramadhani2, M. A. Ndondo1, S. D. Kumwimba1
1Université de Lubumbashi, Faculté des Sciences, Département de Mathématiques et Informatique,

Democratic Republic of the Congo
christopheromak2014@gmail.com, apondondo@gmail.com, didierkumwimba@gmail.com

2Université de Kinshasa, Faculté des Sciences et Technologies, Département de Mathématiques,
Informatique et Statistiques, Democratic Republic of the Congo

issaramadhani@gmail.com
∗Correspondence: christopheromak2014@gmail.com

Abstract. Stochastic differential equations (SDEs) are very often used as models for a large numberof phenomena in the physical, economic and management sciences. They generalize the notion ofordinary differential equations, taking into account a white additive and multiplicative noise term, tomodel random trajectories such as stock market prices or particles movements, on the quantum scale,subject to diffusion phenomena. In rare cases, it is generally impossible to have explicit solutionto these equations. In this case, the numerical approach, presenting itself under various aspects, isthe only favorable outcome. However, the stability of numerical schemes for stochastic differentialequations solution is much more significant. In this paper, we establish and make a classical proofof the mean and mean-square stabilities of the numerical SDEs schemes for Vasicek and GeometricBrownian motion models.

1. Introduction
Stochastic differential equations (SDEs) can be seen as ordinary differential equations, or asintegral equations in which integrals occur with respect to Brownian motion. They were presentedby Ito, with the aim of building continuous and strongly Markovian processes whose generatorsare second-order differential operators called diffusions [6]. In general, solving explicitly stochasticdifferential equations (SDEs), except for cases where the diffusion and drift coefficients are linears,seems difficult or impossible [8]. This is why the numerical approach is relevant because there arenumerical methods allowing to predict the qualitative behavior such as the stability of the solutions.
Received: 20 Jun 2022.
Key words and phrases. Brownian motion; stochastic differential equations; stabilities of SDEs; numerical schemes;vasicek and geometric brownian motion. 1

https://adac.ee
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Eur. J. Math. Anal. 10.28924/ada/ma.3.8 2
The choice of a suitable numerical scheme is based on the understanding and manipulation ofcertain qualitative properties as stability, consistency etc. The qualitative property like stability ofstochastic differential equations solutions, introduced by I.Kats and N.Krasovskii [2] and perfectedby I.I. Gikhman, A.V. Skorokhold [3] and A. Friedman [4] plays a major role in the study of SDEs andthe numerical schemes associated. Thus, looking for numerical schemes that preserve qualitativeproperties as the stability of solutions constitutes and remains a very widespread problem innumerical analysis of SDEs.
In this article we establish and prove the conditions of numerical schemes stabilities in Mean andMean-square. We apply the approach described by Y.Saito [5] to defined and demonstrate the sta-bilities of numericals SDEs schemes as: Euler-Maruyama, Milshtein and Implicit Euler-Maruyamafor Vasicek and geometric Brownian motion models. To begin, let present some elementary notionsrelative to SDEs and the numerical schemes adapted to the SDEs.

2. Preliminary notions
2.1. Stochastic differential equation and stabilities. In this section,we present some definitions inconnection with stochastic differential equation and stabilities of solutions of SDEs.
Definition 2.1. (Stochastic differential equation (SDE) [13]) Let

(
Ω,F, (Ft)t≥0 ,P

)
be a filtered

probability space, (Bt)t≥0 a standard Brownian motion on R
d defines in a filtered probability

space. A stochastic differential equation (SDE) on Rd with the drift coefficient:

b (t,Xt) ∈ [0,T ] ×Rn −→Rn

and the diffusion:

σ (t,Xt) ∈ [0,T ] ×Rn −→Rn×d

when Xo is random variable independent of (Bt)t≥0 is an equation of the form:{
dXt = b (t,Xt) dt + σ (t,Xt) dBt
X (o) = Xo

(2.1)
The white noise σ (t,Xt) can be additive or multiplicative, depending on whether it does notinfluence or does influence the state of the system.

Theorem 2.1. (Existence and uniqueness [14]) We assume that there is a positive constant K
such that ∀ t ≥ 0, X,Y ∈Rd

(1) Lipschitz condition:
|b (t,X) −b (t,Y ) | + |σ (t,X) −σ (t,Y ) | ≤ K|X −Y |

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Eur. J. Math. Anal. 10.28924/ada/ma.3.8 3
(2) Linear growth condition:

|b (t,X) | ≤ K (1 + |X|) ,

|σ (t,X) | ≤ K (1 + |X|)

So the SDE (2.1) admits, for any initial condition Xo of square integrable (E [|Xo|2] < ∞) the
strong solution (Xt)t∈[0,T ],unique, almost surely continuous and satisfying the following condition:

E

(
Sup

0≤t≤T
|X2t |

)
< ∞

Definition 2.2. (Asymptotic stability in probability in large sense [1], [24]) The solution is said to
be asymptotically and stochastically stable in the large sense if

∀ Xo ∈L2Ft ([−T, 0] ,R
n) ,

then

P
{

lim
t−→∞

X (t) = 0
}

= 1.

Definition 2.3. (Stability of pth moment [23], [25])

(1) Let p ≥ 2 we say that a solution of (2.1) is stable in pth moment if ∀� > 0 it exists δ > 0
such as

E

[
Sup
t>0
|X (t) |p

]
< � avec |Xo| < δ

(2) Let p ≥ 2, we say that a solution of (2.1) is stable asymptoticaly in pth moment if it is
stable from peme moment

∀ Xo ∈L2Fto ([−T, 0] ,R
n)

then we have :

lim
T−→∞

E

[
Sup
t>T

|X (t) |p
]

= 0

2.2. Stochastic numerical schemes. In this section we present three numerical schemes as Euler-Maruyama, Implicit Euler-Maruyama and Milshtein schemes.
Definition 2.4. ( Euler-Maruyama scheme [10] , [11]) Let {Xt} the diffusion solution of the SDE(2.1). Let consider the interval [0,T ] and a regular subdivision

t0 = 0 < t1 < t2 < t0 < · · · < tk = T

with step ∆t = T
N

= T
k

, the Euler-Maruyama scheme of (2.1) is defined like:{
XEMk+1 = Xk + b(tk,Xk)(tk+1 − tk) + σ(tk,Xk)(Bk+1 −Bk)
X(0) = X0

(2.2)

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Eur. J. Math. Anal. 10.28924/ada/ma.3.8 4
Definition 2.5. (Implicit Euler-Maruyama scheme [10]) The implicit Euler-Maruyama scheme is
a convergent scheme like the Euler-Maruyama scheme. To be reassured of the existence of the
solutions of this scheme, only the term of the drift is implicit.

For this fact: b(Xk)∆tk which is in the Euler-Maruyama scheme is replaced by b(Xk+1)∆tk
and the diffusion term: σ(Xk)∆Bk remains unchanged.

The implicit Euler-Maruyama scheme of the EDS (2.1) has given by:
XIEMk+1 = Xk + b(Xk+1)∆tk + σ(Xk)∆Bk (2.3)

Definition 2.6. (Milshtein scheme [7]) Let consider the SDE (2.1) and a regular subdivision of the
intervalle et une subdivision of the interval [0,T ]:

0 = t0 < t1 < t2 < · · · < tn = T de [0,T ]

The Milshtein scheme is defined like: XMk+1 = Xk + b(Xk)∆tk + σ(Xk)∆Bk +
1

2
σ(Xk)σ

′(Xk)(∆Bk − ∆tk)

X(0) = X0

(2.4)
Remark 2.1. It should be noted that the Euler-Maruyama scheme converges strongly up to the
order

1

2
while that of Milshtein converges up to the order 1.

3. Numerical stabilities of Vasicek model
3.1. Explicit solution. The Vasicek model (1977) is one of the first stochastic interest rate models.It is a Gaussian process generalizing the Ornstein-Unlenbeck model and explains the observedempirical mean reversion effect on interest rate curves [15], This model looks like:dXt = (θ1 −θ2Xt)dt + θ3dBt

X(0) = X0

∀θ1,θ2 et θ3 > 0 (3.1)
With Xt: the instant interest rate; θ2: mean reversion rate; θ1: the long-term average and θ3: thevolatility.The analytical solution of (3.1) model is:

Xt =
θ1
θ2

+

(
X0 −

θ1
θ2

)
e−θ2t + θ3

∫ +∞
0

e−θ2(t−u)dBu (3.2)
The model (3.1) is equivalent to the model:dXt = θ(µ−Xt)dt + σdBt

X(0) = X0

(3.3)

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Eur. J. Math. Anal. 10.28924/ada/ma.3.8 5
The solution of (3.3) has given by:

Xt = µ + (X0 −µ) e−θt + θ
∫ t

0

e−θ2(t−u)dBu (3.4)
Considering the solution of (3.2), the mean and the mean-square give respectively:

E[Xt] =
θ1
θ2

∀ θ2 > 0

and
V (Xt) =

θ23
2θ2

∀ θ2 > 0

Which means that the stochastic process Xt 'N (θ1θ2 , θ232θ2 )
By using some properties of Brownian motion, the solution of the model (3.2) can be written asfollows:

Xt =
θ1
θ2

+
θ3e
−2θ2t
√

2θ2
B(e2θ2t) (3.5)

Now, we present some numerical stabilities conditions for the system (3.1) of some numericalschemes (Euler-Maruyama, Implicit Euler-Maruyama and Milshtein) and the proofs of these basedon the approach described in [5].
3.2. Euler-Maruyama scheme stabilities. The Euler-Maruyama scheme associated to the system(3.1) is :

XEMk+1 = Xk + (θ1 −θ2Xk) ∆t + θ3∆Bk

XEMk+1 = θ1∆t + (1 −θ2∆t)Xk + θ3
√

∆tZk (3.6)
3.2.1. Mean stability of Euler-Maruyama scheme.
Theorem 3.1. (Mean stability of Euler-Maruyama scheme) The Euler-Maruyama scheme (3.6)
of the Vasicek model (3.1) is Mean asymptotically stable if:

E
[
XEMk+1

]
= (1 −θ2∆t)k+1E[X0] + θ1∆t

[
k+1∑
i=0

(1 −θ2∆t)i
] (3.7)

with |1 −θ2∆t| < 1 and

lim
∆t→0

(
lim

k→+∞
E
[
XEMk+1

])
=
θ1
θ2

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Eur. J. Math. Anal. 10.28924/ada/ma.3.8 6
Proof. To prove the theorem, we start by evaluating the mean of the (3.1) equation using theapproach defined in [5]. In effect,
E
[
XEMk+1

]
= E

[
θ1∆t + Xk (1 −θ2∆t) + θ3

√
∆tZk

]
= E [θ1∆t] + E [Xk (1 −θ2∆t)] + E

[
θ3
√

∆tZk

]
= E [θ1∆t] + (1 −θ2∆t) E [Xk] + 0] with Zk 'N(0, 1)
= θ1∆t + (1 −θ2∆t) E[Xk]

= θ1∆t + (1 −θ2∆t){(1 −θ2∆t) E[Xk−1] + θ1∆t}

= θ1∆t + θ1∆t(1 −θ2∆t) + (1 −θ2∆t)2E[Xk−1]

= θ1∆t(1 + (1 −θ2∆t)) + (1 −θ2∆t)2E[Xk−1]

= θ1∆t(1 + (1 −θ2∆t)) + (1 −θ2∆t)2 {(1 −θ2∆t) E[Xk−2] + θ1∆t}

= θ1∆t(1 + (1 −θ2∆t)) + θ1∆(1 −θ2∆t)2 + (1 −θ2∆t)3E[Xk−2]

= θ1∆t(1 + (1 −θ2∆t) + (1 −θ2∆t)2) + (1 −θ2∆t)3E[Xk−2]

= θ1∆t(1 + (1 −θ2∆t) + (1 −θ2∆t)2 + · · · + (1 −θ2∆t))k+1 + (1 −θ2∆t)k+1E[X0]

= (1 −θ2∆t)k+1E[X0] + θ1∆t

[
k+1∑
i=0

(1 −θ2∆t)i
]

Using the theory of geometric sequences and series, we get:
E
[
XEMk+1

]
= (1 −θ2∆t)k+1E[X0] + θ1∆t

(
(1 − (1 −θ2∆t)k+1)

1 − (1 −θ2∆t)

) (3.8)
As the identity (3.8) represents a geometric sequence, we have that it converges if

|1 −θ2∆t| < 1

By calculating the limit of the (3.8), for ∆t → 0 and k → +∞, we get:
lim

∆t→0

(
lim

k→+∞
E
[
XEMk+1

])
=
θ1
θ2

�

3.2.2. Mean-square stability of Euler-Maruyama scheme.
Theorem 3.2. (Mean-square stability of Euler-Maruyama scheme) The Euler-Maruyama scheme(3.6) of the Vasicek model (3.1) is mean-square asymptotically stable if:

E
[∣∣XEMk+1∣∣2] = (1 −θ2∆t)2(k+1) E (|X0|2) + (θ23 + θ21 ∆t) ∆t k+1∑

i=0

(1 −θ2∆t)2i

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Eur. J. Math. Anal. 10.28924/ada/ma.3.8 7
and that the following two conditions are satisfied simultaneously:

(1) |1 −θ2∆t| < 1

(2) lim
∆t→0

(
lim

k→+∞
E
[
XEMk+1

]2)
=

θ23
2θ2

Proof. As in the previous theorem, we start by calculating the expression:
E
[∣∣XEMk+1∣∣2]

of Vasicek model of the equation (3.1). In effect,
E
[∣∣XEMk+1∣∣2] = |θ1∆t|2 + E (|Xk(1 −θ2∆t)|)2 + θ23 ∆t Zk 'N(0, 1)

= (1 −θ2∆t)2E
(
|Xk|

2
)

+ (θ21 ∆t + θ
2
3 )∆t

= (1 −θ2∆t)2E
(
|Xk|

2
){

E
(
|Xk+1|

2
)

(1 −θ2∆t)2 + (θ23 + θ
2
1 ∆t)∆t

}
+ (θ23 + θ

2
1 ∆t)∆t

= (1 −θ2∆t)4E
(
|Xk−1|

2
)

+ (θ23 + θ
2
1 ∆t)∆t

[
(1 −θ2∆t)2 + 1

]
= (1 −θ2∆t)6E

(
|Xk−2|

2
)

+ (θ23 + θ
2
1 ∆t)∆t[(1 −θ2∆t)

4 + (1 −θ2∆t)2 + 1]

= (1 −θ2∆t)8E
(
|Xk−3|

2
)

+ (θ23 + θ
2
1 ∆t)∆t

[
(1 −θ2∆t)

6
+ (1 −θ2∆t)4 + (1 −θ2∆t)2 + 1

]
= (1 −θ2∆t)2k+1E

(
|X0|2

)
+
(
θ23 + θ

2
1 ∆t

)
∆t
[
(1 −θ2∆t)2k + ... + (1 −θ2∆t)4

+(1 −θ2∆t)2 + (1 −θ2∆t)0
]

= (1 −θ2∆t)
2(k+1)

E
(
|X0|2

)
+
(
θ23 + θ

2
1 ∆t

)
∆t

k+1∑
i=0

(1 −θ2∆t)2i

= (1 −θ2∆t)
2k+2

E
(
|X0|2

)
+
(
θ23 + θ

2
1 ∆t

)
∆t

[
1 −|1 −θ2∆t|

2k+2

1 −|1 −θ2∆t|
2

]

We get:
E
[∣∣XEMk+1∣∣2] = (1 −θ2∆t)2k+2 E (|X0|2) + (θ23 + θ21 ∆t) ∆t [ 1

1 −|1 −θ2∆t|2

] (3.9)
The expression (3.9) as the geometric sequence, we have that it converges when

|1 −θ2∆t| < 1

Passing to the limit of the equation (3.9), for ∆t → 0 and k → +∞, we find the desired result i.e:
lim

∆t→0

(
lim

k→+∞
E
[
XEMk+1

])
=

θ23
2θ2

�

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Eur. J. Math. Anal. 10.28924/ada/ma.3.8 8
3.3. Milshtein’s scheme stabilities. The Milshtein scheme associated to the system (3.1) is:

XMk+1 = θ1∆t + (1 −θ2∆t)Xk + θ3
√

∆tZk (3.10)
with

σ = θ3 σ
′ = 0

Then mean and mean-square stabilities gives the same results as in the Euler-Maruyama schemei.e
Theorem 3.3. (Mean stability of Milshtein scheme) The Milshtein scheme (3.10) of Vasicek model(3.1) is Mean asymptotically stable if:

E
[
XMk+1

]
= θ1∆t

[
k+1∑
i=0

(1 −θ2∆t)i
]

+ (1 −θ2∆t)k+1E[X0]

and that the following two conditions are satisfied simultaneously:

(1) |1 −θ2∆t| < 1

(2) lim
∆t→0

(
lim

k→+∞
E
[
XMk+1

])
=
θ1
θ2

Theorem 3.4. (Mean-square stability of Milshtein scheme)
The Milshtein scheme (3.10) of Vasicek model (3.1) is mean-square asymptotically stable if:

E
[∣∣XMk+1∣∣2] = (θ23 + θ21 ∆t) ∆t k+1∑

i=0

(1 −θ2∆t)2i + (1 −θ2∆t)2(k+1) E
(
|X0|2

)
and that the following two conditions are satisfied simultaneously:

(1) |1 −θ2∆t| < 1

(2) lim
∆t→0

(
lim

k→+∞
E
[
XMk+1

]2)
=

θ23
2θ2

Proof. The proofs of these theorems above is done in the same way as the result theorems of theEuler-Maruyama scheme for Vasicek model. �
3.4. Implicit Euler-Maruyama scheme stabilities. The implicit Euler-Maruyama scheme associ-ated to the system (3.1) is :

XIEMk+1 = Xk + (θ1 −θ2Xk+1)∆t + θ3
√

∆tZk

Xk+1 + θ2∆tXk+1 = Xk + θ1∆t + θ3
√

∆tZk

Xk+1 (1 + θ2∆t) = θ1∆t + Xk + θ3
√

∆tZk

Xk+1 =
θ1∆t

1 + θ2∆t
+

1

1 + θ2∆t
Xk +

θ3
√

∆t

1 + θ2∆t
Zk

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Eur. J. Math. Anal. 10.28924/ada/ma.3.8 9
We get:

XIEMk+1 =
θ1∆t

1 + θ2∆t
+

1

1 + θ2∆t
Xk +

θ3
√

∆t

1 + θ2∆t
Zk (3.11)

3.4.1. Mean stability of implicit Euler-Maruyama scheme.
Theorem 3.5. (Mean stability of implicit Euler-Maruyama scheme) The implicit Euler-Maruyama
scheme (3.11) of Vasicek model (3.1) is Mean asymptotically stable if:

E
(
XIEMk+1

)
=

(
1

1 + θ2∆t

)k+1
E (X0) + θ1∆t

k+1∑
i=0

(
1

1 + θ2∆t

)i
and that the following two conditions are satisfied simultaneously:

(1) |1 + θ2∆t| > 1

(2) lim
∆t→0

(
lim
k→∞

E
[
XIEMk+1

])
=
θ1
θ2

Proof. We start by evaluating the mean of the Implicit Euler-Maruyama scheme of the expressiondefined in (3.11), In effect:
E
(
XIEMk+1

)
= E

(
θ1∆t

1 + θ2∆t

)
+ E

(
1

1 + θ2∆t
Xk

)
+ E

(
θ3
√

∆t

1 + θ2∆t
Zk

)
=

θ1∆t

1 + θ2∆t
+

1

1 + θ2∆t
E (Xk)

=
θ1∆t

1 + θ2∆t
+

1

1 + θ2∆t

{(
1

1 + θ2∆t

)
E (Xk−1) +

θ1∆t

1 + θ2∆t

}
=

θ1∆t

1 + θ2∆t
+

θ1∆t

(1 + θ2∆t)
2

+
1

(1 + θ2∆t)
2
E (Xk−1)

=

(
1

1 + θ2∆t

)2
E (Xk−1) + θ1∆t

(
1

(1 + θ2∆t)
2

+
1

(1 + θ2∆t)

)
=

(
1

1 + θ2∆t

)3
E (Xk−2) + θ1∆t

((
1

1 + θ2∆t

)3
+

(
1

1 + θ2∆t

)2
+

(
1

1 + θ2∆t

))

=

(
1

1 + θ2∆t

)4
E (Xk−3) + θ1∆t

((
1

1 + θ2∆t

)4
+

(
1

1 + θ2∆t

)3
+ · · · + 1

)
By continuing the iterations until k + 1, we obtain:

E
(
XIEMk+1

)
=

(
1

1 + θ2∆t

)k+1
E (X0) + θ1∆t

k+1∑
i=0

(
1

1 + θ2∆t

)i (3.12)
The equation (3.12) is the geometric sum of geometric sequence and geometric series, the expres-sion: (

1

1 + θ2∆t

)
< 1

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Eur. J. Math. Anal. 10.28924/ada/ma.3.8 10
or

|1 + θ2∆t| > 1By using limit of (3.12), for ∆t → 0 and k → +∞, we get:
lim

∆t→0

(
lim

k→+∞
E
[
XIEMk+1

])
=
θ1
θ2

�

3.4.2. Mean-square stability of Implicit Euler-Maruyama scheme.
Theorem 3.6. (Mean-square stability of Implicit Euler-Maruyama scheme) The Implicit Euler-
Maruyama of Vasicek model (3.1) is mean-square asymptotically stable if:

E
(∣∣XIEMk+1 ∣∣2) = ( 11 + θ2∆t

)2(k+1)
E (|X0|)2 +

(
θ23 + θ

2
1 ∆t

)
∆t

k+1∑
i=0

(
1

1 + θ2∆t

)2i
, with |1 + θ2∆t| > 1 and

lim
∆t→0

(
lim
k→∞

E
(
|Xk+1|

2
))

=
θ23

2θ2

Proof. Let us evaluate the mean-square of the implicit Euler-Maruyama scheme (3.11), in effect:
E
(∣∣XIEMk+1 ∣∣2) = E

(∣∣∣∣ θ1∆t1 + θ2∆t
∣∣∣∣2
)

+ E

(∣∣∣∣ 11 + θ2∆tXk
∣∣∣∣) + ∣∣∣∣ θ23 ∆t1 + θ2∆t

∣∣∣∣2
=

θ21 (∆t)
2

(1 + θ2∆t)
2

+
1

(1 + θ2∆t)
2
E
(
|Xk|

2
)

+
θ23 ∆t

(1 + θ2∆t)
2

=
θ21 (∆t)

2 + θ23 ∆t

(1 + θ2∆t)
2

+
1

(1 + θ2∆t)
2
E
(
|Xk|

2
)

=

(
θ23 + θ

2
1 ∆t

)
∆t

(1 + θ2∆t)
2

+

(
1

1 + θ2∆t

)2 {(
1

1 + θ2∆t

)2
E (|Xk−1|)

2
+

(
θ23 + θ

2
1 ∆t

)
∆t

(1 + θ2∆t)
2

}

=

(
1

1 + θ2∆t

)4
E (|Xk−1|)

2
+
(
θ23 + θ

2
1 ∆t

)
∆t

[(
1

1 + θ2∆t

)4
+

(
1

1 + θ2∆t

)2]
...
=

(
1

1 + θ2∆t

)2(k+1)
E (|X0|)2 +

(
θ23 + θ

2
1 ∆t

)
∆t

k+1∑
i=1

(
1

1 + θ2∆t

)2i
By using the geometrical sequence and geometrical series, we get:

E
(∣∣XIEMk+1 ∣∣2) = ( 11 + θ2∆t

)2(k+1)
E (|X0|)2 +

(
θ23 + θ

2
1 ∆t

)
∆t

k+1∑
i=1

(
1

1 + θ2∆t

)2i
we have the geometric sequence and series, converging when |1 + θ2∆t| > 1, by calculating limitof the equation below , for ∆t → 0 and k → +∞, we get:

lim
∆t→0

(
lim

k→+∞
E
[
XIEMk+1

])
=

θ23
2θ2

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Eur. J. Math. Anal. 10.28924/ada/ma.3.8 11
�

4. Numerical stabilities of geometric Brownian motion
4.1. Explicit solution of the model. Geometric Brownian motion known as exponential Brownianmotion is a continuous stochastic process whose logarithm follows a Brownian motion. it is appliedin the mathematical modeling of certain courses in the financial markets [26]. It represents areasonable approximation of the evolution of stock market prices, because a quantity which followsa geometric Brownian motion takes all strictly positive values and only the elementary changesundergone by the random variable are significant.

The geometric Brownian motion Xt is a process which is written in the form [13]:{
dXt = θ1Xtdt + θ2XtdBt

X(0) = X0 ∀θ1,θ2 ∈R
(4.1)

This process admits as an explicit solution:
Xt = X0e

{(θ1−12 θ22 )t+θ2Bt} (4.2)
The variable on the right hand follows a normal distribution, it can also be written in the form:

Xt = Xse
{(θ1−12 θ22 )t+θ2(Bt−Bs )} (4.3)

the conditionnal mean is:
E (Xt|Xs) = Xseθ1(t−s) (4.4)The (4.3) process is often widely used to model the price of a financial asset the return on theasset between two dates is measured by the difference in the logarithms of the prices and is givenby the Gaussian variable below:{

θ1 −
1

2
θ22

}
(t − s) + θ2 (Bt −Bs)

The mean and the mean-square give respectively:
E(Xt) = X0e

θ1t E(X2t ) = X
2
0e

(2θ1+θ22 )t (4.5)
Remark 4.1. It should be noted that:(1) For mean if t →∞ and θ1 < 0 we have:

lim
t→∞

E(Xt) = lim
k→∞

X0e
θ1t = 0

(2) For Mean-square if (2θ1 + θ22) < 0 and t →∞ i.e
lim
t→∞

E(X2t ) = 0 with
(

2θ1 + θ
2
2

)
< 0.

Now, let’s analyze the stabilities of some numerical schemes (Euler-Maruyama, Milshtein andImplicit Euler-Maruyama) in mean and mean-square.

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Eur. J. Math. Anal. 10.28924/ada/ma.3.8 12
4.2. Euler-Maruyama scheme stabilities. The Euler-Maruyama scheme associated to (4.1) is:

XEMk+1 = Xk + θ1Xk∆t + θ2Xk∆Bk

XEMk+1 = Xk (1 + θ1∆t) + θ2Xk
√

∆tZk (4.6)
4.2.1. Mean stability of Euler-Maruyama scheme.
Theorem 4.1. (Mean stability of Euler-Maruyama scheme) The Euler-Maruyama scheme (4.6)
associated to (4.1) model is mean asymptotically stable if

E
[
XEMk+1

]
= (1 + θ2∆t)

k+1E (X0)

with |1 + θ1∆t| < 1 and

lim
∆t→0

(
lim

k→+∞
E
[
XEMk+1

])
= 0

Proof. By calculating the mean of the expression(4.6), we obtain:
E
[
XEMk+1

]
= E

[
Xt(1 + θ1∆t) + θ2Xt

√
∆tZt

]
= E [Xt(1 + θ1∆t)] + E

[
θ2
√

∆tXtZt

]
= E [(1 + θ1∆t)Xt] + E

[
θ2
√

∆t
]
E [Xt] E [Zt]

as Zk 'N(0, 1) E(Zk) = 0
E [Xk+1] = (1 + θ1∆t)E(Xt)

= (1 + θ1∆t) ((1 + θ2∆t)E (Xk−1))

= (1 + θ2∆t)
2E (Xk−1)

= (1 + θ1∆t)
2 ((1 + θ2∆t)E (Xk−2))...

= (1 + θ2∆t)
k+1E (X0)

We get the following geometric sequence:
E
[
XEMk+1

]
= (1 + θ2∆t)

k+1E (X0)

which converges if |1 + θ2∆t| < 1 et nd passing to the limit for a ∆t → 0 and k → +∞, we obtain:
lim

∆t→0

(
lim

k→+∞
E
[
XEMk+1

])
= 0

�

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Eur. J. Math. Anal. 10.28924/ada/ma.3.8 13
4.2.2. Mean-square stability of Euler-Maruyama scheme.
Theorem 4.2. (Mean-square stability of Euler-Maruyama scheme) The Euler-Maruyama scheme(4.6) associated to (4.1) model is mean-square asymptotically stable if:

E
(∣∣XEMk+1∣∣2) = (|(1 + θ1∆t)|2 + ∣∣∣θ2√∆t∣∣∣2)2k+2 E (|X0|2) ,

with
∣∣∣|1 + θ1∆t|2 + ∣∣θ2√∆t∣∣2∣∣∣ < 1 and

lim
∆t→0

(
lim
k→∞

E
(
|Xk+1|

2
))

= 0

Proof. The mean-square of the expression (4.6) gave:
E
[∣∣XEMk+1∣∣2] = E [∣∣∣Xk(1 + θ1∆t) + θ2Xk√∆tZk∣∣∣2]

= E

[
|Xk(1 + θ1∆t)|

2
+
∣∣∣θ2√∆tXkZk∣∣∣2 + 2 ∣∣∣Xk(1 + θ1∆t)θ2√∆tXkZk∣∣∣]

= E
[
|Xk(1 + θ1∆t)|

2
]

+ E

[∣∣∣θ2√∆tXkZk∣∣∣2] + 2E [|Xk(1 + θ1∆t)| ∣∣∣θ2√∆tXkZk∣∣∣]
= |(1 + θ1∆t)|2 E

[
|Xk|

2
]

+
∣∣∣θ2√∆t∣∣∣2 E [|Xk|2]

=

(
|(1 + θ1∆t)|2 +

∣∣∣θ2√∆t∣∣∣2)E [|Xt|2]
=

(
|(1 + θ1∆t)|2 +

∣∣∣θ2√∆t∣∣∣2)(|(1 + θ1∆t)|2 + ∣∣∣θ2√∆t∣∣∣2)E [|Xk−1|2]
...
=

(
|(1 + θ1∆t)|2 +

∣∣∣θ2√∆t∣∣∣2)2k+2 E [|X0|2]
We get a geometric sequence:

E
(∣∣XEMk+1∣∣2) = (|(1 + θ1∆t)|2 + ∣∣∣θ2√∆t∣∣∣2)2(k+1) E (|X0|2)

For ∣∣∣|1 + θ1∆t|2 + ∣∣θ2√∆t∣∣2∣∣∣ < 1 the sequence converges, and passing to the limit, we obtain fora ∀∆t → 0 and k → +∞,
lim

∆t→0

(
lim
k→∞

E
(
|Xk+1|

2
))

= 0

�

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Eur. J. Math. Anal. 10.28924/ada/ma.3.8 14
4.3. Milshtein’s scheme stabilities. The Milshtein Schema associated to the expression(4.1) isgiven by :

XMk+1 = Xk + b(Xk)∆t + σ(Xk)∆Bk +
1

2
σσ′(Xk)

{
(∆Bk)

2 − ∆t
}

= Xk + θ1Xk∆t + θ2Xk∆Bk +
1

2
θ2Xtθ2

{
(∆Bk)

2 − ∆t
}

= Xk + θ1Xk∆t + θ2Xk
√

∆tZk +
1

2
θ22Xk

(
∆tZ2k − ∆t

)
=

(
1 + θ1∆t −

1

2
θ22 ∆t

)
Xk + θ2Xk

√
∆tZk +

1

2
θ22Xk∆tZ

2
k

=

(
1 +

(
θ1 −

1

2
θ22

)
∆t

)
Xk + θ2Xk

√
∆tZk +

1

2
θ22Xk∆tZ

2
kWe have after calculation:

XMk+1 = Xk

(
1 +

(
θ1 −

1

2
θ22

)
∆t

)
+ θ2Xk

√
∆tZk +

1

2
θ22Xk∆tZ

2
k (4.7)

We now consider the same model of geometric Brownian motion, we state some results on thestabilities following Milshtein’s scheme and we prove these results.
4.3.1. Mean stability of Milshtein’s scheme.
Theorem 4.3. (Mean stability of Milshtein’s scheme) The Milshtein’s scheme (4.7) assocated to(4.1) model is mean asymptotically stable if

E
(
XMk+1

)
= [1 + θ1∆t]

k+1
E (X0)

with
|1 + θ1∆t| < 1

and
lim

∆t→0

(
lim
k→∞

E
(
XMk+1

))
= 0

Proof. Applying the usual approach, let us evaluate the mean of gives:
E
(
XMk+1

)
= E

(
Xk

(
1 +

(
θ1 −

1

2
θ22

)
∆t

)
+ θ2Xk

√
∆tZk +

1

2
θ22Xk∆tZ

2
k

)
= E

(
Xk

(
1 +

(
θ1 −

1

2
θ22

)
∆t

))
+ E

(
θ2Xk

√
∆tZk

)
+ E

(
1

2
θ22Xk∆tZ

2
k

)
=

(
1 +

(
θ1 −

1

2
θ22

)
∆t +

1

2
θ22 ∆t

)
E (Xk)

= (1 + θ1∆t) E (Xk)...
= (1 + θ1∆t)

k+1
E (X0)

https://doi.org/10.28924/ada/ma.3.8


Eur. J. Math. Anal. 10.28924/ada/ma.3.8 15
ultimately we get that:

E
(
XMk+1

)
= [1 + θ1∆t]

k+1
E (X0)As the previous expression has the form of a geometric sequence, we know that it converges if

|1 + θ1∆t| < 1, passing to the limit, for all ∆t → 0 and k → +∞ we find the results searched i.e:
lim

∆t→0

(
lim
k→∞

E
(
XMk+1

))
= 0

�

4.3.2. Mean-square stability of Milshtein’s scheme.
Theorem 4.4. (Mean-square stability of Milshtein’s scheme) The Milshtein scheme (4.7) associ-
ated to (4.1) model is mean-square asymptotically stable if

E
(∣∣XMk+1∣∣) =

[∣∣∣∣1 + (θ1 − 12θ22
)

∆t

∣∣∣∣2 + ∣∣∣θ2√∆t∣∣∣2 + ∣∣∣∣12θ22 ∆t
∣∣∣∣2
]2(k+1)

E
(
|X0|2

)
with

∣∣∣∣∣1 + (θ1 − 12θ22) ∆t∣∣2 + ∣∣θ2√∆t∣∣2 + ∣∣12θ22 ∆t∣∣2∣∣∣ < 1 and
lim

∆t→0

(
lim
k→∞

E
(∣∣XMk+1∣∣2)) = 0

Proof. Let’s start by calculating the mean-sqaure of the model expression, ie:
E
(∣∣XMk+1∣∣2) = E

(∣∣∣∣Xk (1 + (θ1 − 12θ22
)

∆t

)
+ θ2Xk

√
∆tZk +

1

2
θ22Xk∆tZ

2
k

∣∣∣∣2
)

= E

(∣∣∣∣Xk (1 + (θ1 − 12θ22
)

∆t

)∣∣∣∣2
)

+ E

(∣∣∣θ2Xk√∆tZk∣∣∣2) + E
(∣∣∣∣12θ22Xk∆tZ2k

∣∣∣∣2
)

=

∣∣∣∣1 + (θ1 − 12θ22
)

∆t

∣∣∣∣2 E (|Xk|2) + ∣∣∣θ2√∆t∣∣∣2 E (|Xk|2) + ∣∣∣∣12θ22 ∆t
∣∣∣∣2 E (|Xk|2)

=

[∣∣∣∣1 + (θ1 − 12θ22
)

∆t

∣∣∣∣2 + ∣∣∣θ2√∆t∣∣∣2 + ∣∣∣∣12θ22 ∆t
∣∣∣∣2
]
E
(
|Xk|

2
)

...
=

[∣∣∣∣1 + (θ1 − 12θ22
)

∆t

∣∣∣∣2 + ∣∣∣θ2√∆t∣∣∣2 + ∣∣∣∣12θ22 ∆t
∣∣∣∣2
]2(k+1)

E
(
|X0|2

)
Continuing with the iterations, we get:

E
(∣∣XMk+1∣∣) =

[∣∣∣∣1 + (θ1 − 12θ22
)

∆t

∣∣∣∣2 + ∣∣∣θ2√∆t∣∣∣2 + ∣∣∣∣12θ22 ∆t
∣∣∣∣2
]2(k+1)

E
(
|X0|2

)
passing to the limit with ∆t → 0 and k → +∞, we obtain the stated results, ie:

lim
∆t→0

(
lim
k→∞

E
(∣∣XMk+1∣∣2)) = 0

�

https://doi.org/10.28924/ada/ma.3.8


Eur. J. Math. Anal. 10.28924/ada/ma.3.8 16
4.4. Implicit Euler-Maruyama scheme stabilities. The implicit Euler-Maruyama scheme (IEM)gives:

XIEMk+1 = Xk + b(Xk+1)∆t + δ(Xt)∆Bk

Xk+1 = Xk + θ1Xk+1∆t + θ2Xt∆Bk

Xk+1 −θ1Xk+1∆t = Xk + θ2Xk∆Bk

Xk+1 (1 −θ1∆t) = Xk + θ2Xk∆BkWe obtain:
XIEMk+1 =

1

1 −θ1∆t
Xk +

θ2
√

∆t

1 −θ1∆t
XkZk Zk 'N(0, 1) (4.8)

4.4.1. Mean stability of implicit Euler-Maruyama scheme.
Theorem 4.5. (Mean stability of implicit Euler-Maruyama scheme) The implicit Euler-Maruyama
scheme (IEM) (4.8) associated to (4.1) model is mean asymptotically stable if

E
(
XIEMk+1

)
=

(
1

1 −θ1∆t

)k+1
E (X0) (4.9)

with |1 −θ1∆t| > 1 then,

lim
∆t→0

(
lim
k→∞

E
(∣∣XIEMk+1 ∣∣2)) = 0

Proof. Let’s evaluate the mean associated to the implicit Euler-Maruyama scheme
E
(
XIEMk+1

)
= E

(
1

1 −θ1∆t
Xk +

θ2
1 −θ1∆t

Xk
√

∆tZk

)
= E

(
1

1 −θ1∆t
Xk

)
+ E

(
θ2

1 −θ1∆t
√

∆t

)
(Xk) (Zk)

= E

(
1

1 −θ1∆t
Xk

)
=

1

1 −θ1∆t
E (Xk)

=

(
1

1 −θ1∆t

)2
E (Xk−1)

=

(
1

1 −θ1∆t

)3
E (Xk−2)

...
=

(
1

1 −θ1∆t

)k+1
E (X0)

Continuing with the iterations we get:
E
(
XIEMk+1

)
=

(
1

1 −θ1∆t

)k+1
E (X0)

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Eur. J. Math. Anal. 10.28924/ada/ma.3.8 17
passing to the limit with ∆t → 0 and k → +∞, we obtain the stated results, ie

lim
∆t→0

(
lim
k→∞

E
(∣∣XIEMk+1 ∣∣2)) = 0

�

4.4.2. Mean-square stability of Implicit Euler-Maruyama scheme.
Theorem 4.6. (Mean-square stability of Implicit Euler-Maruyama scheme) The implicit Euler-
Maruyama scheme associated to the model (4.1) is asymptotically mean-square stable if

E
(∣∣XIEMk+1 ∣∣2) =

[
1 +

∣∣θ2√∆t∣∣
1 −θ1∆t

]2(k+1)
E
(
|X0|2

)
with

∣∣∣∣1+|θ2√∆t|1−θ1∆t
∣∣∣∣ < 1 and

lim
∆t→0

(
lim
k→∞

E
(∣∣XEMIk+1 ∣∣2)) = 0

Proof. : Let us calculate the quadratic mean, in effect,
E
(∣∣XMk+1∣∣2) = E

∣∣∣∣∣ 11 −θ1∆tXt + θ2
√

∆t

1 −θ1∆t
XtZt

∣∣∣∣∣
2


=

(
1

1 −θ1∆t

)2
E

(∣∣∣Xk + θ2√∆tXkZk∣∣∣2)
=

(
1

1 −θ1∆t

)2 [
E
(
|Xk|

2
)

+ E

(∣∣∣+θ2√∆tXkZk∣∣∣2)]
=

∣∣∣∣ 11 −θ1∆t
∣∣∣∣2 [E (|Xk|2) + ∣∣∣+θ2√∆t∣∣∣2 E (|Xk|2)]

=

(
1 +

∣∣θ2√∆t∣∣2)
(1 −θ1∆t)2

E
(
|Xk|

2
)

=

(
1 +

∣∣θ2√∆t∣∣2)
(1 −θ1∆t)2

(
1 +

∣∣θ2√∆t∣∣2)
(1 −θ1∆t)2

E
(
|Xk−1|

2
)

Continuing with the iterations, we get:
E
(∣∣XEMIk+1 ∣∣2) =

[
1 +

∣∣θ2√∆t∣∣
1 −θ1∆t

]2(k+1)
E
(
|X0|2

)
passing to the limit with ∆t → 0 and k → +∞, we obtain the stated results, i.e :

lim
∆t→0

(
lim
k→∞

E
(∣∣XMk+1∣∣2)) = 0

�

https://doi.org/10.28924/ada/ma.3.8


Eur. J. Math. Anal. 10.28924/ada/ma.3.8 18
5. Numerical simulations and residual calculations

In this section, we present some numerical simulations for Vasicek and geometric Brownianmotion models using Matlab and we calculate the errors between the exact solution and thatobtained by applying the numerical schemes of Euler-Maruyama, Milshtein and Implicit Euler-Maruyama.
5.1. Numerical simulation of Vasicek and geometric Brownian motion models. We present somesimulations of Vasicek and Brownian geometric motion models in the increasing and decreasingcases.

Figure 1. Increasing Vasicek model

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Eur. J. Math. Anal. 10.28924/ada/ma.3.8 19
Figure 2. Decreasing Vasicek model

Figure 3. Increasing geometric Brownian motion model

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Eur. J. Math. Anal. 10.28924/ada/ma.3.8 20
Figure 4. Decreasing geometric Brownian motion model

5.2. Interpretation of results.
5.2.1. Vasicek model. The figures 1 and 2 show the stability of the Vasicek model in the increasingand decreasing cases, in these figures we see that the Euler-Maruyama scheme coincides with thatof Milshtein. We have in the first two figures of figure 1 the following errors: Emerr = 0.2280,Milerr = 0.2280 and Iemerr = 0.2258 In both figures of figure 1 Emerr = 0.3007, Milerr = 0.3007and Iemerr = 0.2851In both figures of figure 2 Emerr = 0.2268, Milerr = 0.2268 and Iemerr = 0.2237.
5.2.2. Geometric Brownian motion model. The figures figure 3 et figure 4 present the stability ofgeometric Brownian motion in the increasing and decreasing cases. Indeed, the first three figuresin figure 3 present the increasing stability of geometric motion and the last figure in figure 3 andthe two figures in figure 4 show the decreasing stability of the model. We have in the first twofigures and figure 3 the following errors: Emerr = 0.0027, Milerr = 0.0011 and Iemerr = 0.0013and for the third figure in figure 3: Emerr = 0.0177, Milerr = 0.0111 and Iemerr = 0.0128. In theboth figures of figure 4: Emerr = 0.0054, Milerr = 0.0022 and Iemerr = 0.0026.
Remark 5.1. From the results bellow, in the cases of increasing and decreasing stabilities of
Vasicek et Geometric Brownian motion, we have that, the Milshtein scheme is the best scheme
because it’s the best approximates the exact solution.

6. Conclusion
We have presented in this article the analysis of the stability in mean and mean-square forVasicek and geometric Brownian motion models. In these models, we established the conditionsof the numerical stabilities of Euler-Maruyama, implicit Euler-Maruyama and Milshtein schemes.these conditions have been proved by using classical manner and Y. Saito’s approach. It should benoted that each case is different from the other depending on whether the models examined have

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Eur. J. Math. Anal. 10.28924/ada/ma.3.8 21
additive (Vasicek model) or multiplicative (Geometric brownian motion) white noise type.

Finally, for these models, we found that the stability conditions of the Vasicek model coincideswith the stability of the ODEs, on the other hand, for the stability conditions of the second modelto coincide with the stability of the ODEs, it is necessary that θ2 < 0. To support these results,numerical simulations were made and the calculations of the residuals (errors) comes in support ofthe results found. In the next work we will analyze the numerical stabilities of these two modelsby using Non-standard Euler-Maruyama scheme.

References
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https://doi.org/10.1142/9789812798886_0026
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https://doi.org/10.1007/978-3-642-23280-0

	1. Introduction
	2. Preliminary notions
	2.1. Stochastic differential equation and stabilities
	2.2. Stochastic numerical schemes

	3. Numerical stabilities of Vasicek model
	3.1. Explicit solution
	3.2. Euler-Maruyama scheme stabilities
	3.3.  Milshtein's scheme stabilities
	3.4. Implicit Euler-Maruyama scheme stabilities

	4. Numerical stabilities of geometric Brownian motion
	4.1. Explicit solution of the model
	4.2. Euler-Maruyama scheme stabilities
	4.3. Milshtein's scheme stabilities
	4.4. Implicit Euler-Maruyama scheme stabilities

	5. Numerical simulations and residual calculations
	5.1. Numerical simulation of Vasicek and geometric Brownian motion models
	5.2. Interpretation of results

	6. Conclusion
	References
	Bibliographie

