EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 12, No. 2, 2019, 448-468
ISSN 1307-5543 – www.ejpam.com
Published by New York Business Global

Existence and Uniqueness Solution under Non-lipschitz
condition of the mixed fractional Heston’s model

Didier Alain Njamen Njomen1,∗, Eric Djeutcha1, Louis-Aimé Fono2

1 Department of Mathematics and Computer’s Science, Faculty of Science, University of
Maroua, Maroua, Cameroon
2 Department of Mathematics and Computer’s Science, Faculty of Science,
University of Douala, Cameroon

Abstract. This paper focuses on a mixed fractional version of Heston model in which the volatility
Brownian and price Brownian are replaced by mixed fractional Brownian motion with the Hurst
parameter H ∈ ( 3

4 , 1) so that the model exhibits the long range dependence. The existence and
uniqueness of solution of mixed fractional Heston model is established under various non-Lipschitz
condition and a related Euler discretization method is discussed. An example on the American
put option price using Least Squares Monte Carlo Algorithm to produce acceptable results under
the mixed fractional Heston model is presented to illustrate the applicability of the theory. The
numerical result obtained proves the performance of our results.

2010 Mathematics Subject Classifications: 60J65, 60G22, 60H07, 91B25, 91B26

Key Words and Phrases: Brownian motion, fractional processes, mixed fractional Brownian,
Heston mode, Monte Carlo Algorithm

1. Introduction

The fractional Brownian motion (fBm) BH = {BH
t , t ∈ [0, T ]} with Hurst parameter

H ∈ (0, 1) is a Gaussian self-similar process with stationary increments. This process
was introduced by [18] and studied by [23], where a stochastic integral representation in
terms of a standard Brownian motion (Bm for short) was established. The parameter H
is called Hurst index from the statistical analysis, developed by the climatologist [17]. The
self-similarity and stationary increments properties make the fBm an appropriate model
for many applications in diverse fields from biology to finance [19], [22]. If H 6= 1

2 , the
process BH

t is neither a semimartingale, nor Markovian process. Therefore, the classical
Ito calculus can not be used to analyze the fBm process. In order to overcome this problem,
we use a mixed fractional Brownian motion (mfBm for short) with a, b and H parameters

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v12i2.3395

Email addresses: didiernjamen1@gmail.com (D. A. N. Njamen),
djeutchaeric@yahoo.fr (E. Djeutcha), lfono2000@yahoo.fr (L-A. Fono)

http://www.ejpam.com 448 c© 2019 EJPAM All rights reserved.



D. A. N. Njamen, E. Djeutcha / Eur. J. Pure Appl. Math, 12 (2) (2019), 448-468 449

which is a linear combination of Brownian motion and fractional Brownian motion with
parameter H, defined for any t ∈ [0, T ] by:

MH
t = {aBt + bBH

t , ∀a, b > 0}, (1)

where BH is the fBm with Hurst parameter H ∈ (0, 1). [10] proved that the mfBm
process with Hurst parameter H ∈]34 , 1[ is equivalent to a martingale aBt and hence it is
arbitrage-free.

Let T > 0 be a fixed time and (Ω, F, (Ft)t∈[0,T ],P) be a given filtered complete proba-
bility space with (Ft)t∈[0,T ] being a filtration that satisfies the usual hypothesis.

The aim of this paper is to study the following stochastic differential equation (SDE
for short) on Rn

dX(t) = µ(t,X(t))dt+ σ(t,X(t))dMH
t . (2)

On most occasions, the coefficients of SDEs driven by mfBm are assumed to satisfy the
Lipschitz condition. The existence and uniqueness of solutions of SDEs driven by fBm with
Lipschitz condition have been studied in [13], [11]. Fortunately, many researchers have
investigated the SDEs under non-Lipschitz condition and they presented many meaningful
results [31], [5], [29, 30]. But, to the best of our knowledge, the existence and uniqueness
of solutions of SDEs driven by mfBm with a non-Lipschitz condition have not been con-
sidered. This point motivates us to carry out the present study. In the present paper, we
discuss the SDEs with mfBm under the non-Lipschitz condition. Using the successive ap-
proximation method, the existence and uniqueness theorems of solutions to the following
non-Lipschitz SDEs driven by mfBm are proved:

X(t) = x0 +

∫ t

0

µ(s,X(s))ds+

∫ t

0

σ1(s,X(s))dBs +

∫ t

0

σ2(s,X(s))dBHs , (3)

where t ∈ [0, T ], x0 = ξ ∈ Rn is a random variable, 0 ≤ T < ∞, the process B represent
a m-dimensional standard Ft-Brownian motion and the process BH represent a d-dimensional
Ft-adapted fractional Brownian motion with the Hurst index H ∈ ( 3

4 , 1) defined in a same com-
plete probability space (Ω,F ,P), and µ(t,X(t))[0, T ] × R → R,σ1(t,X(t)) : [0, T ] × R → R and
σ2(t,X(t)) : [0, T ]× R→ R are all mesurable functions.

The main difficulty when considering Equation (3) lies in the fact that both stochastic integrals
are dealt in different ways. However, the integral with respect to the Bm is an It integral, while
the integral with respect to the fBm has to be understood in the pathwise sense. Finally, using the
LSM Algorithm, we will calculate the value of the American put option price under the Heston
model governed by MH ( mixed fractional Heston model (in short MFH)) for differents values of
the Hurst parameter H and we compare the result with the value of American option price under
Heston Model (HM). We remind that in (3),

∫ t
0
·dBs stand for the stochastic integral w.r.t Bm,∫ t

0
·dBHs stand for the stochastic integral w.r.t fBm.
The article is organized as follows. In section 2, we recall briefly the malliavin calculus in order

to define the integral with respect to fBm and introduce proper normed spaces and we also state
our assumptions on the coefficients µ, σ1 and σ2 of Equation (3).

In section 3, we give a version of Heston’s model which the volatility Brownian and stock
price Brownian are replaced by the mixed fractional Brownian motion. The main existence and
uniqueness results are discussed under the non-Lipschitz condition and simulation result using the



D. A. N. Njamen, E. Djeutcha / Eur. J. Pure Appl. Math, 12 (2) (2019), 448-468 450

discritization of Euler’s method for 80 simulation paths of the stock price is provided. Finally, in
section 4, we give the applicability of the general theory to calculate the value of American put
option price under the MFH model and we compare the result with the value of American put
option price under the Heston Model (HM).

2. Preliminaries

The (elementary) background needed here about the theory of integral with respect fBm [26],
[20]. Let (Ω,F ,P) be a complete filtered probability space satisfying the usual assumptions. We
fix some notation throughout the paper | · | will denote the absolute value of a real number, the
Euclidean norm of a vector, or the operator norm of a matrix. The symbol K will denote a generic
constant, whose value may change from one value to another. � denote the Wick product and is
defined in [7]. If A is a vector or matrix, its transpose is denoted by AT .

Stochastic differential equation with respect to fBm have been interpreted via various stochastic
integral, such as the Wick-integral, the Wiener integral, the Skorohod integral, and path-wise
integral [26], [14],[2, 3], [9], [25]. In this paper, we consider the path-wise integral with respect to
fBm [26].

2.1. Stochastic integral with respect to fBm.

We begin by a brief review of the malliavin calculus. We start with the definition of the integral
with respect to fBm as a path-wise stochastic integrals (symmetric, forward and backward integral)
for fBm on the slow-fast systems, following the work of [7].

Definition 1. Let u(t) be a stochastic process with integrable trajectories.

(i) The symmetric integral of u(t) with respect to BHt is defined as

lim
ε→0

1

2ε

∫ T

0

u(s)
[
BHs+ε −BHs−ε

]
ds, (4)

provided that the limit exists in probability, and is denoted by
∫ T
0
u(s)d◦BHs .

(ii) The forward integral of u(t) with respect to BHt is defined as

lim
ε→0

1

ε

∫ T

0

u(s)

[
BHs+ε −BHs

ε

]
ds, (5)

provided that the limit exists in probability, and is denoted by
∫ T
0
u(s)d−BHs .

(iii) The backward integral of u(t) with respect to BHt is defined as

lim
ε→0

1

ε

∫ T

0

u(s)

[
BHs−ε −BHs

ε

]
ds, (6)

provided that the limit exists in probability, and is denoted by
∫ T
0
u(s)d+BHs .

For the convenience of readers, some basic properties of the path-wise stochastic integrals are
provided as follows:

Let ϕ : R+ × R+ → R+ be defined by

ϕ(t, s) = H(2H − 1)|t− s|2H−2, t, s ∈ R+, (7)



D. A. N. Njamen, E. Djeutcha / Eur. J. Pure Appl. Math, 12 (2) (2019), 448-468 451

where H is a constant with 1
2 < H < 1.

Let g : R+ → R be borel mesurable, define

L2
ϕ(R+) =

{
g : ‖g‖2ϕ =

∫
R+

∫
R+

g(t)g(s)ϕ(t, s)dsdt <∞

}
, (8)

if we equip L2
ϕ(R+) and with the inner product

< g1, g2 >ϕ=

∫
R+

∫
R+

g1(t)g2(s)ϕ(t, s)dsdt, g1, g2 ∈ R+ (9)

then L2
ϕ(R+) become respectively the separable Hilbert space.

Let S be the set of smooth and cyndrical random variables of the form

F (ω) = f

(∫ T

0

ψ1(t)dBHt , . . . ,

∫ T

0

ψn(t)dBHt

)
(10)

where n ≥ 1, f ∈ C∞b (Rn+)(f and all its partial derivatives are bounded) and ψi ∈ H.
H is the completion of the mesurable functions such that ‖ψi‖2 <∞ and ψn, is the sequence

in H such that < ψi, ψj >ϕ= δij . We denote by H the space of measurable functions h on [0, T ]
satisfying

‖h‖2H =

∫ T

0

∫ T

0

|h(t)||h(s)|ϕ(t, s)dsdt <∞, (11)

where H is a Banach space with the norm ‖ · ‖2H.
The malliavin derivative DH

t of a smooth and cylindrical random variable F ∈ S is defined as
the H-value random variable:

DH
t F =

n∑
i=1

∂f

∂xi

(∫ T

0

ψ1(t)dBHt , . . . ,

∫ T

0

ψn(t)dBHt

)
ψi(t), (12)

then for any p ≥ 1, the derivative operator DH
t is a closable operator from Lp(Ω) into Lp(Ω,H).

We define the ϕ-derivative of F :

Dϕ
t F =

∫
R+

ϕ(t, v)DH
v Fdv. (13)

Definition 2. The space Lϕ[0, T ] of integrands is defined as the family of stochastic process u(t) on

[0, T ], such that E
∫ T
0
‖u(s)‖2ϕ <∞ u(t) is ϕ-differentiable, the trace of Dϕ

t u(t) exists; 0 ≤ s ≤ T,

0 ≤ t ≤ T, E
∫ T
0

∫ T
0

[Dϕ
s u(s)]2dsdt < ∞ and for each sequence of partition (πn, n ∈ N) such that

|πn| → 0 as n→ 0

n−1∑
i=0

E
∫ t

(n)
i+1

t
(n)
i

∫ t
(n)
j+1

t
(n)
j

∣∣∣Dϕ
s u

π(t
(n)
i )uπ(t

(n)
j )−Dϕ

s u(t)Dϕ
t u(s)

∣∣∣ dsdt
and E[‖uπ − u‖2ϕ]2 tend to 0 as n→ 0, where πn = t

(n)
0 < t

(n)
1 < . . . < t

(n)
n−1 < t

(n)
n = T.

According to the remark 1 in [3] and Proposition 6.2.3 in [7]. Let u(t) be a stochastic process

in the space D1,2(|H|), and satisfies
∫ T
0

∫ T
0

∣∣DH
s u(s)

∣∣2H−2 dsdt <∞ then we can see that symetric

integral
∫ T
0
u(s)d◦BHs coincides with the forward and backward integrals.



D. A. N. Njamen, E. Djeutcha / Eur. J. Pure Appl. Math, 12 (2) (2019), 448-468 452

If u(s) ∈ Lϕ[0, T ], then one of the pathwise integrals exist and the following relation holds :∫ T

0

u(s)d◦BHs =

∫ T

0

u(s) � dBHs +

∫ T

0

Dϕ
t u(s)ds. (14)

Lemma 1. Let BHt be the fBm with 1
2 < H < 1 and u(t) be a stochastic process in D1,2(|H|) ∩

(Lϕ[0, T ]), then for every T <∞,

E

[∫ T

0

u(s)d◦BHs

]2
≤ 2HT 2H−1E

[∫ T

0

|u(s)|2ds

]
+ 4TE

[∫ T

0

Dϕ
s u(s)

]2
ds. (15)

Proof. We have:

E

[∫ T

0

u(s)d◦BHs

]2
= E

[∫ T

0

u(s) � dBHs +

∫ T

0

Dϕ
t u(s)ds

]2

≤ 2E

[∫ T

0

u(s) � dBHs

]
+ 2E

[∫ T

0

Dϕ
s u(s)

]2
ds

≤ 2HT 2H−1E

[∫ T

0

|u(s)|2ds

]
+ 4TE

[∫ T

0

Dϕ
s u(s)

]2
ds.

2.2. Hypothesis of non-Lipschitz condition

Throughout this paper we assume that the coefficients µ, σ1 and σ2, which are continuous,
satisfy, for all x, y ∈ Rn and t ∈ [0, T ], the assumptions (A.1) and (A.2):

A.1 The functions µ and σ1 have a linear growth and satisfy suitable modulus of continuity with
respect to variable x uniformly in t. Assumption A.1 means that µ and σ1 satisfy:

A.1.1 µ(t, x)| ≤ K(1 + |x|)
A.1.2 |µ(t, x)− µ(t, y)|2 ≤ %(|x− y|2)

A.1.3 σ1(t, x)| ≤ K(1 + |x|)
A.1.4 σ1(t, x)− σ1(t, y) ≤ %(|x− y|2),

where % is a concave increasing function from R+ to R+ such that %(0) = 0,
%(u) > 0 for u > 0 and ∫

0+

du

%(u)
= +∞. (16)

A.2 The functions µ(t, 0) and σ2(t, 0) are locally integral with respect to t, and the function σ2
is continuously differentiable in the first variable t. Assumption A.2 means that µ and σ2
satisfy ∀t ∈ [0, T ], µ(t, .), σ2(t, .) ∈ Lϕ([0, T ]) ∩ D1,2(|H|):

E |µ̃(t, x, y)|2 + E |σ̃2(t, x, y)|2 + E|Dϕ
t (σ̃2(t, x, y))|2 ≤ %

(
E|x− y|2

)
, (17)

with ϕ is given by (7),and {
σ̃2(t, x, y) = σ2(t, x)− σ2(t, y)
µ̃(t, x, y) = µ(t, x)− µ(t, y).

(18)

The non-Lipschitz condition has a variety of forms [4], [27], [1] and [28].



D. A. N. Njamen, E. Djeutcha / Eur. J. Pure Appl. Math, 12 (2) (2019), 448-468 453

Now, let define two sequences of functions {χn(t)}n=1,2,... and {χ̃n,p(t)}n=1,2,... such that χ1(t) =

Ct, χn+1(t) =
∫ t
0
%1(χn(s))ds and χ̃n,p(t) = sup

0≤s≤t
E |Xn+p(s)−Xn(s)|2 , n = 1, 2, . . . . where p ≥ 1

is fixed arbitrarily.

Lemma 2. (Liu, [20]) Under the non-Lipschitz condition,

0 ≤ χ̃n,p(t) ≤ χn(t) ≤ χn−1(t) ≤ . . . ≤ χ1(t), (19)

for all positive integer n.

Lemma 3. There exists a positive number G,

∀µ(t, ·), σ2(t, ·) ∈ Lϕ([0, T ]) ∩ D1,2(|H|),

E |µ(t, x)|2 + E |σ2(t, x)|2 + E |Dϕ
t (σ2(t, x))|2 ≤ G

(
1 + E|x|2

)
. (20)

Proof. Since %(u) is a concave and non-negative function, we can choose two positive constants
a > 0 and b > 0, so that κ(u) ≤ a+ bu

E|µ(t, x)|2 + E|σ2(t, x)|2 + E|Dϕ
t (σ2(t, x))|2

≤ 2E
(
|µ(t, 0)|2 + |σ2(t, 0)|2 + |Dϕ

t (σ2(t, 0))|2
)

+ 2E|µ(t, x)− µ(t, 0)|2 + 2E|σ2(t, x)− σ2(t, 0)|2 + 2E|Dϕ
t (σ2(t, x))− σ2(t, 0)|2

≤ 2 sup
0≤t≤T

E
(
|µ(t, 0)|2 + |σ2(t, 0)|2 + |Dϕ

t (σ2(t, 0))|2
)

+ 2%(E(x)2)

≤ G(1 + E(x)2),

where
G = 2 sup

0≤t≤T

{
E
(
|µ(t, 0)|2 + |σ2(t, 0)|2 + |Dϕ

t (σ2(t, 0))|2
)

+ 2a, 2b
}
<∞.

3. The main results

3.1. The MFH model framework

Mixed Fractional Heston model is the Heston model in which the volatility Brownian and the
price Brownian are replaced by the MFBM. So we first consider the Heston model which will be
described in Definition 3.

Definition 3. ( Heston Model, [15]) The model given by [15] as one of the most the important
stochastic volatility models. In this model the volatility is a stochastic process and it is determined
by the stochastic differential equation (SDE) as follows, see [21]

dSt = Stµdt+
√
VtStdB1,t (21)

dVt = κ(θ − Vt)dt+ σ
√
VtdB2,t (22)

dB1,t × dB2,t = ρdt, (23)

where B1,t and B2,t are two Brownian motion process with correlation ρ ∈ (−1, 1) and S represent
the current stock price, V is the volatility, κ is the rate which V reverts to θ, θ is the long variance
and σ is the volatility of the volatility.



D. A. N. Njamen, E. Djeutcha / Eur. J. Pure Appl. Math, 12 (2) (2019), 448-468 454

Definition 4. ( MFH model) Let us consider a probability space (Ω, F,P) on some Brownian
motion Bi = Bi,t, fractional Brownian motion BHi = BHi,t, for i = 1, 2. Let (Ft)t≥0 be a filtration
generated by these three above process and P a risk neutral probability under the asset price process
St at time t ≥ 0. Let Vt be stochastic volatility process at time t ≥ 0. In the Heston model, if we
substitute Bi,t by MH

i = MH
i,t then, we obtain a Mixed Fractional Heston model and its SDE’s is

given by {
dSt = Stµdt+

√
VtStdM

H
1,t

dVt = κ(θ − Vt)dt+ σ
√
VtdM

H
2,t

(24)

with dMH
1,t × dMH

2,t = ρ(a2dt+ b2dt2H), ρ ∈ (0, 1). i.e.

dSt = Stµdt+ a
√
VtStdBt,1 + b

√
VtStdB

H
1,t (25)

and

dVt = κ(θ − Vt)dt+ aσ
√
VtStdBt,2 + bσ

√
VtStdB

H
2,t, (26)

where κ control the speed of mean reversion of the volatility and θ is the long-run mean of the
volatility, σ is the volatility of Vt process. S0 and V0 are spot asset price and spot variance respec-
tively.

Vt is strictly positive when 2κθ ≥ σ2 and non-negative when 0 ≤ 2κθ < σ2(Feller condition).
ρ is the coefficient of correlation between Bi, fractional Brownian motion BHi .

3.2. Simulation of MFH model

Euler’s scheme is the simplest way to discritize the stochastic differential equations [16]. We
perform Euler discretisation on the MFH model. The Euler discretization can be used to approx-
imate the asset path of the stock price on a discrete time grid [16]. Let St be an asset price which
implies in (25) and Vt satisfying (26). Let Λ = {t0, t1, . . . , tN} be a partition of the interval [0, T ].
i.e 0 < t0 < t1 < . . . < tN = T then, we have for all 0 ≤ j ≤ N − 1 and i = 1, 2,

Sj+1 = Sj + µSj∆t+
√
VjSj∆M

H
ij (27)

Vj+1 = Vj + κ(θ − Vj)∆t+ σ
√
Vj∆M

H
ij . (28)

We have the following formula, for all 0 ≤ j ≤ N − 1 , tj = j∆t and i = 1, 2, 3.

∆MH
ij = MH

i (tj+1)−MH
i (tj); (29)

and
∆MH

ij ∼ N
(
0, a2∆t+ b2∆t2H

)
. (30)

According to the central limit theorem, we have

∆MH
ij = N

(
0, aZi

√
∆t+ bZi

√
∆t2H

)
(31)

and
Zi ∼ N (0, 1).

Therefore, we have

Sj+1 = Sj + µSj∆t+ a
√
Vj∆tSjZ1 + bSjZ1

√
Vj∆t2H , (32)



D. A. N. Njamen, E. Djeutcha / Eur. J. Pure Appl. Math, 12 (2) (2019), 448-468 455

and

Vj+1 = Vj + κ(θ − Vj)∆t+ aσ
√
Vj∆tφ1 + bσφ1

√
Vj∆t2H , (33)

where the correlated normal variables, φ1 = ρZ1+
√

1− ρ2Z2 generated by the Cholesky’s method.
The parameters of the MFH model are taken from [24] and are presented in the following table

1. The asset price has been estimated under the MFH model, where the parameter of the option
model is given by the table 1 defined by:

T ρ S0 V0 µ σ κ θ ∆t E
2 0.26 0.04 0.04 0.07 0.04 2 3 0.001 100

Table 1: Parameter of MFH model

(a) [H=0.76] (b) [H=0.77] (c) [H=0.78]

Figure 1: Simulated asset paths of MFH model.

In Figure 1 above, we see 80 simulated paths for the asset price with different Hurst parameter
H such that H > 3

4 . Figure 1 below shows that increasing or decreasing the hurst parameters
affects the future price of the asset so that, by increasing the Hurst parameter, the difference
between expected lowest price and the highest price will be increased.

The simulation of the MFH model is given by the following algorithm

Algorithm 1. MFH model simulation process.

(i) Set ∆ = t
N .

(ii) For i = 1 to number of simulation.

(iii) Generate independent standard normal variables, Zj ∼ N (0, 1), j = 1, . . . , N .

(iv) SetSj+1 ← Sj + µSj∆t+ a(Vj∆t)
1
2SjZ1 + bSjZ1(Vj∆t

2H)
1
2 .

(v) For Vj+1 ← Vj + κ(θ − Vj)∆t+ aσ(Vj∆t)
1
2φ1 + bσφ1(Vj∆t

2H)
1
2 .

(vi) End For.



D. A. N. Njamen, E. Djeutcha / Eur. J. Pure Appl. Math, 12 (2) (2019), 448-468 456

3.3. Existence and Uniqueness

Now we will discuss the solutions for non-Lipschitz SDE’s with Brownian motion, fBm for each
equation defined in (58) by using an iteration of Picard [12]. Let X0(t) = ζ be a random variable
with E|ζ|2 < +∞. In the general case, we construct an approximative sequence of stochastic
process {Xp(t)}p≥1 as follows

Xp(t) = ζ +

∫ t

0

µ(s,Xp−1(s))ds+

∫ t

0

σ1(s,Xp−1(s))dBs

+

∫ t

0

σ2(s,Xp−1(s))dBHs . (34)

Theorem 1. Under the assumptions A.1 and A.2, the path-wise uniqueness holds for (3), t ∈
[0, T ].

Proof. Let Y (t) and Z(t) be two solutions of (34) and Y (0) = Z(0), we have

Y (t)− Z(t) =

∫ t

0

µ̃(s)ds+

∫ t

0

σ̃1(s)dBs +

∫ t

0

σ̃2(s)d◦BHs

= Iµ(t) + Iσ1
(t) + Iσ2

(t)

with 
Iµ(t) =

∫ t
0
µ̃(s)ds

Iσ1
(t) =

∫ t
0
σ̃1(s)dBs

Iσ2
(t) =

∫ t
0
σ̃2(s)d◦BHs .

(35)

where  µ̃(s) = µ(s, Y (s))− µ(s, Z(s))
σ̃1(s) = σ1(s, Y (s))− σ1(s, Z(s))
σ̃2(s) = σ2(s, Y (s))− σ2(s, Z(s)),

(36)

By employing the following inequality

∀a1, a2, a3 ∈ R, |a1 + a2 + a3|2 ≤ 3|a1|2 + 3|a2|2 + 3|a3|2. (37)

It follows that

|Y (t)− Z(t)|2 ≤ 3

∣∣∣∣∫ t

0

µ̃(s)ds

∣∣∣∣2 + 3

∣∣∣∣∫ t

0

σ̃1(s)dBs

∣∣∣∣2 + 3

∣∣∣∣∫ t

0

σ̃2(s)d◦BHs

∣∣∣∣2 . (38)

We observe

E|Y (t)− Z(t)|2 ≤ 3E|Iµ(t)|2 + 3E|Iσ1
(t)|2 + 3E|Iσ2

(t)|2. (39)

We have to estimate E|Iµ(t)|2, E|Iσ1
(t)|2 and E|Iσ2

(t)|2.
According to the Ito’s Isometry we have

E
∣∣∣∣∫ t

0

σ̃1(s)dBs

∣∣∣∣2 = E
∫ t

0

|σ̃1(s)|2 ds. (40)

By using Fubini’s Theorem, we have

E
∫ t

0

|σ̃1(s)|2 ds =

∫ t

0

E |σ̃1(s)|2 ds. (41)



D. A. N. Njamen, E. Djeutcha / Eur. J. Pure Appl. Math, 12 (2) (2019), 448-468 457

Using the linear growth assumption in (A.1.4), it is easy to see that

|σ̃1(s)|2 ≤ %(|Y (s)− Z(s)|2) (42)

and then
E |σ̃1(s)|2 ≤ E[%(|Y (s)− Z(s)|2)]. (43)

According to the Jensen’s Inequality, we have

E
[
%
(
|Y (s)− Z(s)|2

)]
≤ %

(
E |Y (s)− Z(s)|2

)
, (44)

it follows that ∫ t

0

E |σ̃1(s)|2 ds ≤
∫ t

0

%
(
E |Y (s)− Z(s)|2

)
ds. (45)

Therefore,

E|Iσ1(t)|2 ≤
∫ t

0

%
(
E |Y (s)− Z(s)|2

)
ds. (46)

Using the simple estimation, we have

E |Iµ(t)|2 = E
∣∣∣∣∫ t

0

µ̃(s)ds

∣∣∣∣2 ≤ TE ∫ t

0

|µ̃(s)|2 ds, (47)

and by Fubini’s Theorem, we have

E |Iµ(t)|2 ≤ T
∫ t

0

E |µ̃(s)|2 ds. (48)

Consequently

E|Iµ(t)|2 ≤ 8T

∫ t

0

E|µ̃(s)|2ds. (49)

We know that BHt is the fBm with 1
2 < H < 1 and σ̃2(t) is a stochastic process in D1,2(|H|) ∩

(Lϕ[0, T ]), for every t ∈ [0, T ], we have from lemma 1 that

E
[∫ t

0

σ̃2(s)d◦BHs

]2
≤ 2Ht2H−1E

[∫ t

0

|σ̃2(s)|2ds
]

+ 4tE
[∫ t

0

Dϕ
s σ̃2(s)

]2
ds. (50)

The inequality (50) implies that

E|Iσ2(t)|2 ≤ 8T

∫ t

0

[
E|σ̃2(s)|2 + E|Dϕ

s σ̃2(s)|2
]
ds. (51)

By combining (51) and (49), we obtain

E|Iµ(t)|2 + E|Iσ2(t)|2 ≤ 8T

∫ t

0

[
E|µ̃(s)|2ds+ E|σ̃2(s)|2 + E|Dϕ

s σ̃2(s)|2
]
ds. (52)

Using the inequality (17) of the assumption (A.2), we obtain

E|Iµ(t)|2 + E |Iσ2
(t)|2 ≤ 8T

∫ t

0

%
(
E|Y (t)− Z(t)|2

)
ds. (53)



D. A. N. Njamen, E. Djeutcha / Eur. J. Pure Appl. Math, 12 (2) (2019), 448-468 458

The inequality (46) and (53) give

E|Iσ1
(t)|2 + E|Iµ(t)|2 + E|Iσ2

(t)|2 ≤ (1 + 8T )

∫ t

0

%(E |Y (s)− Z(s)|2)ds. (54)

The inequality (39) give

E|Y (t)− Z(t)|2 ≤ (3 + 24T )

∫ t

0

%(E |Y (s)− Z(s)|2)ds. (55)

Noticing that from (16), the inequality (55) implies that E|Y (t) − Z(t)|2 = 0,∀t ∈ [0, T ]. Since
T > 0 is an arbitrary, Y (t) ≡ Z(t), ∀t ∈ [0, T ]. Thus the path-wise uniqueness holds for (3).

To prove the existence of Theorem 1, we show that under the non-Lipschitz condition,

lim
n,i→∞

sup
0≤t≤T

E|Xp(t)−Xi(t)|2 = 0. (56)

We call {Xp(·)}p≥1 a Cauchy sequence which X(·) is its limit. By letting p → ∞ in (34), we
deduce that the solution to (1) exist. We fix p ≥ 1 arbitrary and define two sequences of functions
{χn(t)}n≥1 and {χ̃n,p(t)}n≥1 where

χ1(t) = Ct

χn+1(t) =
∫ t
0
%1(χn(s))ds

χ̃n,p(t) = sup
0≤t≤T

E|Xp(t)−Xi(t)|2.
(57)

By Lemma 2, we observe that (χn(t)) decreases when n → ∞ and is nonnegative function on
t ∈ [0, T ], therefore, we define χ(t) as limit of (χn(t)), we have χ(0) = 0 and χ(t) is a continuous
function on t ∈ [0, T ]. Or χ(t) = lim

n→∞
χn(t), we have

lim
n→∞

χn+1(t) = lim
n→∞

∫ t

0

%1(χn(s))ds =

∫ t

0

%1(χ(s))ds. (58)

Since χ(0) = 0 and
∫
0+

du
%(u) = +∞, we say that (58) implies χ(t) = 0, therefore we get from the

inequality (19)

0 ≤ lim
n,p→∞

χn+1(t) = lim
n→∞

sup
0≤t≤T

E|Xp(t)−Xi(t)|2

= lim
n,p→∞

χ̃n,p(T )

≤ lim
n→∞

χ̃n(T ) = 0, (59)

namely, lim
n,i→∞

sup
0≤t≤T

E|Xn(t)−Xi(t)|2 = 0.

Remark 1. The asset price St satisfy

dSt = Stµdt+ a
√
VtStdBt,1 + b

√
VtStdB

H
1,t. (60)

With respect to the above equation defined by (60), we can say that the stock price equation of the
MFH model has a unique solution.



D. A. N. Njamen, E. Djeutcha / Eur. J. Pure Appl. Math, 12 (2) (2019), 448-468 459

Let V0(t) = ζ be a random variable with E|ζ|2 < +∞.
We also construct an approximative sequence of stochastic process {V nt }n≥1 as follows

V nt = ζ +

∫ t

0

κ(θ − V n−1s )ds+

∫ t

0

σa
√
V n−1s dBs

+

∫ t

0

σb
√
V n−1s dBHs . (61)

Theorem 2. Under the assumptions A.1 and A.2, the volatility of the mixed Heston model has
a unique positive solution Vt, where t ∈ [0, T ] and T = inf{t > 0|Xt = 0}.

Proof. We have

Vt = ζ +

∫ t

0

κ(θ − Vs)ds+

∫ t

0

σ
√
VsdM

H
s . (62)

Suppose that for some initial value ζ there are two continous solutions Vt and Ṽt satisfy (62), then
the difference satisfies

Vt − Ṽt =

∫ t

0

ψ1(s)ds+ σa

∫ t

0

ψ2(s)dBs + σb

∫ t

0

ψ3(s)dBHs

= J1(t) + J2(t) + J3(t) (63)

where 
J1(t) =

∫ t
0
ψ1(s)ds

J2(t) = σa
∫ t
0
ψ2(s)dBs

J3(t) = σb
∫ t
0
ψ3(s)dBHs

(64)

with 
ψ1(t) = −κ(Vs − Ṽs)ds

ψ2(t) = σa(
√
Vs −

√
Ṽs)

ψ3(t) = σb(
√
Vs −

√
Ṽs).

(65)

We observe that
E|Vt − Ṽt|2 ≤ 3E|J1(t)|2 + 3E|J2(t)|2 + 3E|J3(t)|2. (66)

Now we have to estimated E|Ji(t)|2, i = 1, 2, 3.
By letting

ε = min

{√
Vt) +

√
Ṽt > 0|t ∈ [0, T ]

}
, (67)

according to the Ito isometry, we get

E|J2(t)|2 = E
∫ t

0

|ψ2(s)|2 ds. (68)

By Fubini’s Theorem

E|J2(t)|2 =

∫ t

0

E |ψ2(s)|2 ds. (69)

Using the assumption in (A.1.2), we have

E |ψ2(s)|2 ≤ σ2a2

ε2
E
(
%
∣∣∣(Vs − Ṽs)∣∣∣2) , (70)



D. A. N. Njamen, E. Djeutcha / Eur. J. Pure Appl. Math, 12 (2) (2019), 448-468 460

and the Jensen’s inequality give

E |ψ2(s)|2 ≤ σ2a2

ε2
%

(
E
∣∣∣(Vs − Ṽs)∣∣∣2) . (71)

Therefore

E|J2(t)|2 ≤ σ2a2

ε2

∫ t

0

%

(
E
∣∣∣(Vs − Ṽs)∣∣∣2) ds, (72)

using the simple estimation and Fubini’s Theorem, we have

E|J1(t)|2 ≤ T
∫ t

0

E |ψ1(s)|2 ds. (73)

Let M = min{ 1
ε2 , 1}, we have

E|J1(t)|2 ≤ 8TM

∫ t

0

E |ψ1(s)|2 ds. (74)

We know that BHt is the fBm with 1
2 < H < 1 and ψ3(t) is a stochastic process in D1,2(|H|) ∩

(Lϕ[0, T ]), for every t ∈ [0, T ], we have from lemma 1 that

E
[∫ t

0

ψ3(s)d◦BHs

]2
≤ 1

ε2

[
2Ht2H−1E

[∫ t

0

|ψ̃3(s)|2ds
]

+ 4tE
[∫ t

0

Dϕ
s ψ̃3(s)

]2
ds

]
(75)

with ψ̃3(s) = σc(Vs − Ṽs).
The inequality (75) implies that

E|J3(t)|2 ≤ 8TM

∫ t

0

[
E|ψ̃3(s)|2 + E|Dϕ

s ψ̃3(s)|2
]
ds. (76)

By combining (76) and (74), we obtain

E|J1(t)|2 + E|J3(t)|2 ≤ 8TM

∫ t

0

[
E|ψ1(s)|2ds+ E|ψ3(s)|2 + E|Dϕ

s ψ3(s)|2
]
ds. (77)

Using the inequality (17) of the assumption (A.2), we obtain

E|J1(t)|2 + E|J2(t)|2 ≤ 8TM

∫ t

0

%
(
E|Vs − Ṽs|2

)
ds. (78)

The inequality (72) and (78) give

E|J1(t)|2 + E|J2(t)|2 + E|J3(t)|2 ≤M(σ2a2 + 8T )

∫ t

0

%

(
E
∣∣∣Vs − Ṽs∣∣∣2) ds. (79)

The inequality (66) give

E|Vt − Ṽt|2 ≤ 3M(σ2a2 + 8T )

∫ t

0

%

(
E
∣∣∣Vs − Ṽs∣∣∣2) ds. (80)

Noticing that from (16), the inequality (80) implies that E|Vt − Ṽt|2 = 0,∀t ∈ [0, T ]. Since T > 0

is an arbitrary, Vt ≡ Ṽt, ∀t ∈ [0, T ]. Thus the path-wise uniqueness holds for (3). To show the
existence, we proceed in the same way as Theorem 1.



D. A. N. Njamen, E. Djeutcha / Eur. J. Pure Appl. Math, 12 (2) (2019), 448-468 461

Theorem 3. If Vt is the solution of (62), then E(|Vt|2) <∞.

Proof. To prove the existence of Theorem 3, we show that under the assumptions A.1 and
A.2,

E|V nt |2 <∞, (81)

where for all n ≥ 1, V nt verify (61). By induction, we have for n = 1, from Lemma 1, and Lemma
15,

E|V 1
t |2 ≤ 3E |ζ|2 + 3E

∣∣∣∣∫ t

0

κ(θ − V 0
s )ds

∣∣∣∣2
+ 3E

∣∣∣∣∫ t

0

σ1(s,
√
V 0
s )dBs

∣∣∣∣2 + 3E
∣∣∣∣∫ t

0

σ2(s,
√
V 0
s )dBHs

∣∣∣∣2
≤ 3E|ζ|2 + 3κ2θ2t2 + 12T

∫ t

0

[
E
∣∣κ(s, V 0

s )
∣∣2 + ξ1

]
ds

with
ξ1 = E

∣∣σ2(s, V 0
s )
∣∣2 + E

∣∣Dϕ
s σ2(s, V 0

s )
∣∣2 . (82)

Let ε = min
0≤t≤T

{
1; 1√

V 0
t

}
, we have

E|V 1
t |2 ≤ 3E|ζ|2 + 3κ2θ2t2 + 12Tε2

∫ t

0

[
E
∣∣κ(s, V 0

s )
∣∣2 + ξ1

]
ds.

Moreover
E
∣∣κ(s, V 0

s )
∣∣2 + ξ1 ≤ G

(
1 + E|V 0

s |2
)
, (83)

with G = max

[
2 sup
0≤t≤T

E(|κV 0
s |2 + ξ1) + 2a1, 2b1

]
, where

ξ1 = |σ2(s, 0)|2 + |Dϕ
s σ2(s, 0)|2 . (84)

Therefore, we get

E|V 0
t |2 ≤ 3E|ζ|2 + 3κ2θ2t2 + 12TGε2

∫ t

0

(
1 + E|V 0

s |2
)
ds. (85)

As we know that V 0
s = ζ, then we have

E|V 1
t |2 ≤ 3E|ζ|2 + 3κ2θ2t2 + 12TGε2t

(
1 + E|ζ|2

)
(86)

i.e
E|V 1

t |2 ≤ 3E|ζ|2 + 3κ2θ2T 2 + 12GT 2ε2
(
1 + E|ζ|2

)
<∞. (87)

The relation (87) is true. Now assume that for all n,

E|V nt |2 ≤ A+ 3E|ζ|2
[

n∑
i=0

(12GT )i

i!
ti +

p∑
i=1

(12GT )i

i!
ti

]
,



D. A. N. Njamen, E. Djeutcha / Eur. J. Pure Appl. Math, 12 (2) (2019), 448-468 462

with A = 3E|ζ|2 + 3κ2θ2T 2 + 12GT 2. Then we have , for n+ 1

E|V n+1
t |2 ≤ 3E|ζ|2 + 3κ2θ2t2 + 12GT

∫ t

0

(
1 + E|V nt |2

)
ds

+ 12GT

∫ t

0

(1 + E|V nt |2)ds

≤ 3E|ζ|2 + 3κ2θ2t2 + 12GT

∫ t

0

(
1 + E|V nt |2

)
ds

≤ A+ 12GT

∫ t

0

(
E|V ns |2

)
ds

≤ A+ 3E|ζ|2
∫ t

0

[
n∑
i=0

(12GT )i+1

i!
si +

p∑
i=1

(12GT )i+1

i!
si

]
ds

≤ A+ 3E|ζ|2
∫ t

0

[
n+1∑
i=1

(12GT )i

i!
si +

p+1∑
i=2

(12GT )i

i!
si

]
ds

≤ A+ 3E|ζ|2
∫ t

0

[
n+1∑
i=1

(12GT )i

i!
si +

p+1∑
i=2

(12GT )i

i!
si

]
ds

≤ A+ 3E|ζ|2
[
n+1∑
i=1

(12GT )i

i!
ti +

p+1∑
i=2

(12GT )i

i!
ti

]
,

with A = A(1 + 12GT 2).
Hence

E|V n+1
t |2 ≤ A(1 + 12GT 2) + 3E|ζ|2

[
n+1∑
i=1

(12GT )i

i!
ti +

p+1∑
i=2

(12GT )i

i!
ti

]
.

Or ex '
n∑
i=1

exp(x), we have

E|V n+1
t |2 ≤ 3E|ζ|2(1 + 12GT 2) + 3κ2θ2T 2(1 + 12GT 2)

+ 12GT 2(1 + 12GT 2) + 3E|ζ|2 exp(12GT 2) <∞.

We can conclude that E|V nt |2 <∞ and therefore E(|Vt|2) <∞.

Remark 2. If St satisfy the stock price of the MFH model, then E|St|2 <∞.

4. American put option under MFH model

In mathematical finance, Monte Carlo simulation methods are best ways for pricing the Amer-
ican option. The benefit of the Monte Carlo simulation method is to trade with dependent options.
This method can simulate the underlying asset price path by path, then obtain the payoff associated
with the data for each simulated path and using the average discounted payoff to approach the ex-
pected discounted payoff which is the value of path dependent option. Least Squares Monte Carlos
method (in short LSM) is more suitable for problems in higher dimensions than oder comparable
Monte Carlos Method [6], [8].



D. A. N. Njamen, E. Djeutcha / Eur. J. Pure Appl. Math, 12 (2) (2019), 448-468 463

4.1. LSM Algorithm
In the LSM approach, in order to have a better price, we only recognize the in-the-money

paths. The LSM algorithm is given as follows,

Algoritheorem 2. LSM algorithm

(i) Set Sj MFH model Asset Path.

(ii) If Sj < E.

(iii) Set cashflow(j) = (E − Sj)e−rδt,j = 1,. . . ,number of paths.

(iv) Else cashflow(j) = 0.

(v) End if.

(vi) For j = N - 1 : -1 : 1.

(vii) Set index =find(E − Sj > 0).

(viii) Set X = [ons(size(index))S(index)(S(index))2].

(ix) Set B = (XTX)
1
2Xcashflow(index).

(x) Set conditional exp = XB.

(xi) If conditional exp < −E − Sj , j = 1 : size(index,1).

(xii) Set cashflow (index(j))= (E − Sj),j = 1:size(index,1).

(xiii) End if.

(xiv) Set cashflow = cashflow e(−rδt).

(xv) End if.

(xvi) Set American put option = mean(cashflow).

we apply the above LSM algorithm for pricing American put option when the underlying stock
price follows the MFH model. The details of LSM Algorithm can be found in [21]. The concept
of the put option is related to stopping time process. Indeed, it can be expired at any time until
expiration date. Let Θ be a set of stopping times and St be a stock price. The price of the
American put option is defined as follows

P (τ, S(τ)) = sup
{
E
[
e−rτ (E − S(τ))

+
]}

, τ ∈ Θ, (88)

where S(0) is initial stock price.
If τ = +∞, then the value of the American put option is zero. Here, we investigate the value

of the American put option of the MFH model by considering different values of the expiration
date and a, b parameter.



D. A. N. Njamen, E. Djeutcha / Eur. J. Pure Appl. Math, 12 (2) (2019), 448-468 464

4.2. Experimental results

In this section, we present some numerical results for the institution the price of the American
put option of the MFH model by using the above LSM algorithm. We first simulate the price of the
action, the paths based on the Euler scheme described in subsection 3.2. We use the parameters
given in the table 1 for the price of the American put option under the MFH model. By applying
the algorithm 1 and 2, we get the following results for differents value of Hurst parameters H such
that H ∈ ( 3

4 , 1]:

a b 2 3
0.1 MFH 49.7453 49.6720
0.2 HM 49.5860 49.6969

Table 2: Comparison of American put option using MFH models as function of
S0 and V0 for H=0.76.

a b 2 3
0.1 MFH 49.7970 49.6887
0.2 HM 49.6450 49.7773

Table 3: Comparison of American put option using MFH models and HM as
function of S0 and V0 for H=0.77.

a b 2 3
0.1 MFH 49.7305 49.7286
0.2 HM 49.5938 49.6614

Table 4: Comparison of American put option using MFH models and HM as
function of S0 and V0 for H=0.78.

a b 2 3
0.1 MFH 49.7298 49.6620
0.2 HM 49.9172 49.7155

Table 5: Comparison of American put option using MFH model and HM as
function of S0 and V0 for H=0.79

In tables 2, 3, 4 and 5 the results are shown for some quantities of Hurst parameters H=0.76, 0.77, 0.78,0.79.
We see that increase in the value of the Hurst parameter leads to a significant increase in the value
of the American put option price under MFH.

a b 2 3
0.1 MFH 49.5432 49.7045
0.2 HM 49.5249 49.7619

Table 6: Comparison of American put option using MFH model and HM as
function of S0 and V0 for H=0.80.



D. A. N. Njamen, E. Djeutcha / Eur. J. Pure Appl. Math, 12 (2) (2019), 448-468 465

a b 2 3
0.1 MFH 49.9421 49.7051
0.2 HM 49.6671 49.6763

Table 7: Comparison of American put option using MFH model and HM as
function of S0 and V0 for H=0.85.

a b 2 3
0.1 MFH 49.8725 49.7520
0.2 HM 49.8213 49.7047

Table 8: Comparison of American put option using MFH model and HM as
function of S0 and V0 for H=0.87.

a b 2 3
0.1 MFH 49.8149 49.7072
0.2 HM 49.6484 49.7865

Table 9: Comparison of American put option using MFH model and HM as
function of S0 and V0 for H=0.89.

In tables 6, 7, 8 and 9, the results are shown for some quantities of Hurst parameters H=0.80,H=0.85, 0.87, 0.89.
We see that increase in the value of the Hurst parameter leads to a significant increase in the value
of the American put option price under MFH.

a b 2 3
0.1 MFH 49.7141 49.6956
0.2 HM 49.6721 49.6781

Table 10: Comparison of American put option using MFH model and HM as
function of S0 and V0 for H=0.90.

a b 2 3
0.1 MFH 49.8439 49.7099
0.2 HM 49.9487 49.6653

Table 11: Comparison of American put option using MFH model and HM as
function of S0 and V0 for H=0.95.

a b 2 3
0.1 MFH 49.7129 49.6898
0.2 HM 49.8712 49.7194

Table 12: Comparison of American put option using MFH model and HM as
function of S0 and V0 for H=0.97.

a b 2 3
0.1 MFH 49.7509 49.6964
0.2 HM 49.7540 49.7373



REFERENCES 466

Table 13: Comparison of American put option using MFH model and HM as
function of S0 and V0 for H=0.99.

In tables 10, 11, 12 and 13, the results are shown for some quantities of Hurst parameters
H=0.90,H=0.95, 0.97, 0.99. We see that increase in the value of the Hurst parameter leads to
a significant increase in the value of the American put option price under MFH.

5. Conclusion

In this paper, we have studied the application of LSM algorithm to estimate the value of
American put option price of the MFH model that both the stock price and volatility in the model
are governed by distinct processes. For this reason, this version of the Heston model that we have
proposed in this paper is intuitive and computational efficient. We have proved that our model has
a unique solution. Moreover, we have used Euler discretization method which performed the MFH
model. Numerical examples showed that the LSM algorithm produces acceptable results which
generalized those of the Heston model.

Acknowledgements

The second author received scholarship for his Doctoral studies from the Non-Governmental
Organization “Jean Felicien Gacha’s Foundation”. And the research work has been done under the
research Grant No 17-497RG/MATHS/AF/ACG−FR3240297728 offered by the World Academy
of Sciences (TWAS) to the Applied Mathematics to Social Sciences Research Group of the Labora-
tory of Mathematics-University of Douala-Cameroon. The authors sincerely thanks Jean Felicien
Gacha’s Foundation and TWAS for their immeasurable help.

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