International Journal of Analysis and Applications
ISSN 2291-8639
Volume 4, Number 2 (2014), 107-121
http://www.etamaths.com

GLOBAL UNIQUENESS RESULT FOR FUNCTIONAL

DIFFERENTIAL EQUATIONS DRIVEN BY A WIENER

PROCESS AND FRACTIONAL BROWNIAN MOTION

TOUFIK GUENDOUZI∗ AND SOUMIA IDRISSI

Abstract. We prove a global existence and uniqueness result for the solu-

tion of a mixed stochastic functional differential equation driven by a Wiener
process and fractional Brownian motion with Hurst index H > 1/2. We also

study the dependence of the solution on the initial condition.

1. Introduction

Fractional Brownian motion (fBm) with a Hurst parameter H ∈ (0, 1) is
defined formally as a continuous centered Gaussian process BHt = {BHt , t ≥ 0}
with the covariance

(1) RH =
1

2
(t2H + s2H −|t−s|2H).

For H > 1/2 it exhibits a property of long-range dependence, which makes it a
popular model for long-range dependence in natural sciences, financial mathematics
etc. For this reason, equations driven by fractional Brownian motion have been an
object of intensive study during the last decade.

From (1) we deduce that IE(|BHt −BHs |2) = |t−s|2H and, as a consequence, the
trajectories of BH are almost surely locally α-Hölder continuous for all α ∈ (0,H).
Since BH is not a semimartingale if H 6= 1/2 (see [7]), we cannot use the classical
Itô theory to construct a stochastic calculus with respect to the fBm. Over the
last years some new techniques have been developed in order to define stochastic
integrals with respect to fBm. Essentially two different types of integrals can be
defined:

One possibility is Skorokhod, or divergence integral introduced in the fractional
Brownian setting in [2]. However this definition is not very practical: it is based
on Wick rather than usual products, and unlike Brownian case, in the fractional
Brownian case this makes difference when integrating non-anticipating functions
because of dependence of increments. This makes this definition worthless for most
applications (most notably, those in financial mathematics). Moreover, it is impos-
sible to solve stochastic differential equations with such integral except the cases of
additive or multiplicative noise; the latter case was considered in [10].

Another approach is a pathwise integral, defined first in [13] for fBm with

2010 Mathematics Subject Classification. 60G15, 60G22, 60H10.
Key words and phrases. Fractional Brownian motion, Wiener process, Mixed stochastic func-

tional differential equation, Fractional integrals and derivatives.

c©2014 Authors retain the copyrights of their papers, and all
open access articles are distributed under the terms of the Creative Commons Attribution License.

107



108 TOUFIK GUENDOUZI AND SOUMIA IDRISSI

H > 1/2 as a Young integral. The papers [6, 11] were the first to prove exis-
tence and uniqueness of stochastic differential equations involving such integrals.
Later the pathwise approach was extended with the help of Lyons?rough path the-
ory to the case of arbitrary H in [1] where also unique solvability of equations with
H > 1/4 was proved.

Very recently, the stochastic differential equations driven simultaneously by a
fractional Brownian motion and standard Brownian motion have been studied by
several authors. In [5] Guerra and Nualart have proved an existence and uniqueness
theorem for solutions of multidimensional, time dependent, stochastic differential
equations driven by a multidimensional fractional Brownian motion with Hurst
parameter H > 1/2 and a multidimensional standard Brownian motion using a
techniques of the classical fractional calculus and the classical Itô stochastic calcu-
lus. Their (existence) result is based on the Yamada-Watanabe theorem. In [8] the
existence and uniqueness of solutions is proved by Mishura and Shevchenko for dif-
ferential equations driven by a fractional Brownian motion with parameter H > 1/2
and a Wiener process in one dimensional case, under mild regularity assumptions
on the coefficients. For the same equation, with nonhomogeneous coefficients and
random initial condition, the convergence in Besov space of the solutions depending
on a parameter has been studied in [9] by Mishura and Posashkova.

In this paper we focus on the following mixed stochastic functional differen-
tial equation involving Wiener process and fractional Brownian motion, with non-
constant delay
(2)

x(t) = φ(0) +

∫ t
0

b(s,xs)ds +

∫ t
0

σW (s,xs)dW(s) +

∫ t
0

σH(s,xs)dB
H(s), t ≥ 0

x0 = φ ∈Cr,

where BH = {BH(t); t ∈ [0,T]} is a fractional Brownian motion with Hurst index
H ∈ ( 1

2
, 1), W = {W(t); t ∈ [0,T]} is a Wiener process and Cr is the space of

all continuous functions f from [−r, 0] to IR endowed by the uniform norm ‖ · ‖.
Here, xt ∈Cr denote the function defined by xt(u) = x(t + u), ∀u ∈ [−r, 0] and the
coefficients b,σW ,σH : [0,T] ×Cr → IR are appropriate functions. The stochastic
integral w.r.t. Wiener process in (2) is the standard Itô integral, and the integral
w.r.t. fBm is pathwise generalized Lebesgue-Stieltjes, or Young integral.

Our goal in this paper is to prove the existence and uniqueness of the solution
for equation (2). Then we will study the dependance of the solution on the initial
condition. We first prove our results for deterministic equations and we will easily
apply them pathwise to the Wiener processus and fractional Brownian motion.

The paper is organized as follows. In Section 2, we state the problem and list
our assumptions on the coefficients of Eq. (2). Section 3, contains some basic facts
about extended Stieltjes integrals. In Section 4, we derive some precise estimates
for the integrals involved in Eq. (2). Section 5 is devoted to obtain the existence,
uniqueness and dependence on the initial data for the solution of the deterministic
equations. In Section 6, we apply the results of the previous sections to stochastic
equations driven by both Wiener process and fractional Brownian motion and we
give the proofs of our main theorems.



GLOBAL UNIQUENESS RESULT 109

2. Main result

Let (Ω,F, (Ft, t ∈ [0,T]),IP) be a complete probability space with a filtration
satisfying the standard conditions. Denote by {W(t),Ft, t ∈ [0,T]} the standard
Wiener process adapted to this filtration. Suppose that B = {B(t); t ∈ [0,T]} is
an Ft-fBm with Hurst index H ∈ ( 12, 1). Consider the mixed stochastic functional
differential equation (2) and let us consider the following assumptions on the coef-
ficients.

(Hb) The function b(t,y) is continuous. Moreover, it is Lipschitz continuous in
the variable y and has linear growth in the same variable, uniformly in t, that is,
there exist constants L1 and L2 such that

|b(t,y) − b(t,z)| ≤ L1‖y −z‖,
|b(t,y)| ≤ L2(1 + ‖y‖),

for all y,z ∈Cr and t ∈ [0,T].

(HσW ) The function σW (t,y) is continuous. Moreover, it is Lipschitz continu-
ous in y and has linear growth in the same variable, uniformly in t, that is, there
exist constants L3 and L4 such that

|σW (t,y) −σW (t,z)| ≤ L3‖y −z‖,
|σW (t,y)| ≤ L4(1 + ‖y‖),

for all y,z ∈Cr and t ∈ [0,T].

(HσH) The function σH(t,y) is continuous and Fréchet differentiable in the variable
y. Moreover, there exist constants L5, L6 and L7 such that

|∇yσH(t,y)|L(Cr,IR) ≤ L5,
|∇yσH(t,y) −∇yσH(t,z)|L(Cr,IR) ≤ L6‖y −z‖,

|σH(t,y) −σH(s,y)| + |∇yσH(t,y) −∇yσH(s,y)|L(Cr,IR) ≤ L7|t−s|,
for all y,z ∈Cr and t ∈ [0,T].
Note that (HσH) implies the linear growth property, i. e., there exists a constant
L such that

|σH(t,y)| ≤ L(1 + ‖y‖),
for all y ∈Cr and t ∈ [0,T].

Let us define for λ ∈ (0, 1] the space Cλ of λ-Hölder continuous functions f :
[0,T] → IR, equipped with the norm

‖f‖λ := ‖f‖∞ + sup
0≤s<t≤T

|f(t) −f(s)|
(t−s)λ

< ∞,

where ‖f‖∞ := sup
t∈[0,T]

|f(t)|.

Our main results are the following theorems on the uniqueness, existence and
dependence of the solution of Eq. (2) on the initial condition.

Theorem 2.1. Let the assumptions (Hb), (HσW ) and (HσH) be satisfied, and C
be a generic constant which depends on the constants Li, 1 ≤ i ≤ 7.

(1) If 1 − H < α < H and φ is a stochastic process whose trajectories belong
to the space C1−α([−r, 0]) IP-a.s, then there exists a unique solution x of
mixed equation (2) with paths in C1−α([−r, 0]) IP-a.s.



110 TOUFIK GUENDOUZI AND SOUMIA IDRISSI

(2) If in addition α + H > 3
2

, C is independent of ω and the process φ satisfies
IE‖φ‖p1−α < ∞ for p ≥ 1, then the solution x satisfies IE‖x‖

p
1−α < ∞ for

p ≥ 1.

Theorem 2.2. Let the assumptions (Hb), (HσW ) and (HσH) be satisfied, φ,φ
n ∈

C1−α([−r, 0]) and C be a generic constant which depends on the constants Li, 1 ≤
i ≤ 7. Let x be a solution of the mixed equation (2) and xn the solution of the same
equation with φn in place of φ. We assume that 1 −H < α < H.

(1) If lim
n
‖φn −φ‖1−α = 0, a.s., then we have, for IP-almost all ω ∈ Ω,

lim
n
‖xn(ω,.) −x(ω,.)‖1−α = 0.

(2) If in addition α+H > 3
2

, C is independent of ω and φ,φn are deterministic
functions, then lim

n
IE‖xn −x‖p1−α = 0 for p ≥ 1.

Remark 2.3. We note that the regularity and absolute continuity results for the
above mixed equation in d-dimensional case, but without delay, was studied in [5]
by Guerra and Nualart. For the equations driven only by fBm, and the constant
delay situation, we refer the reader to [4].

3. Generalized Stieltjes integral

Let α ∈ (0, 1
2
). For any measurable function f : [0,T] → IR we introduce the

following notation

(3) ‖f(t)‖α := |f(t)| +
∫ t
0

|f(t) −f(s)|
(t−s)α+1

ds.

Denote by Wα,∞ the space of measurable functions f : [0,T] → IR such that

(4) ‖f(t)‖α,∞ := sup
t∈[0,T]

‖f(t)‖α < ∞.

A equivalent norm can be defined by

(5) ‖f‖α,µ = sup
t∈[0,T]

e−µt
(
|f(t)| +

∫ t
0

|f(t) −f(s)|
(t−s)α+1

ds

)
; µ ≥ 0.

Note that for any �, (0 < � < α), we have the inclusions

Cα+�([0,T]; IR) ⊂ Wα,∞([0,T]; IR) ⊂Cα−�([0,T]; IR) (for more details, see [7]).

In particular, both the fractional Brownian motion BH, with H > 1
2
, and the

standard Brownian motion W , have their trajectories in Wα,∞. We refer the reader
to [7, 5] for further details on this topics.

We denote by W
1−α,∞
T ([0,T]; IR) the space of continuous functions g : [0,T] → IR

such that

‖g‖1−α,∞,T := sup
0<s<t<T

(
|g(t) −g(s)|
(t−s)1−α

+

∫ t
s

|g(y) −g(s)|
(y −s)2−α

dy

)
< ∞.

Clearly, for all � > 0 we have

C1−α+�([0,T]; IR) ⊂ W1−α,∞T ([0,T]; IR) ⊂C
1−α([0,T]; IR).

Denoting

Λα(g; [0,T]) =
1

Γ(1 −α)
sup

0<s<t<T
|(D1−α

t−
gt− )(s)|,



GLOBAL UNIQUENESS RESULT 111

where Γ(α) =

∫ ∞
0

rα−1e−rdr is the Euler function and

(D1−α
t−

gt− )(s) =
eiπ(1−α)

Γ(α)

(
g(s) −g(t)
(t−s)1−α

+ (1 −α)
∫ t
s

g(s) −g(y)
(y −s)2−α

dy

)
1(0,t)(s).

We also define the space Wα,1([0,T]; IR) of measurable functions f on [0,T] such
that

‖f‖α,1;[0,T] =
∫ T
0

[
|f(t)|
tα

+

∫ t
0

|f(t) −f(y)|
(t−y)α+1

dy

]
dt < ∞.

We have Wα,∞([0,T]; IR) ⊂ Wα,1([0,T]; IR) and ‖f‖α,1;[0,T] ≤
(
T + T

1−α

1−α

)
‖f‖α,∞;[0,T].

In [13], Zähle introduced the generalized Stieltjes integral

(6)

∫ T
0

f(t)dg(t) = (−1)α
∫ T
0

(Dα0+f)(t)(D
1−α
T− gT−)(t)dt,

defined in terms of the fractional derivative operators

(Dα0+f)(t) =
1

Γ(1 −α)

(
f(t)

tα
+ α

∫ t
0

f(t) −f(y)
(t−y)α+1

dy

)
1(0,T)(t),

and

(D1−αT− gT−)(t) =
e−iπα

Γ(α)

(
gT−(t)

(T − t)1−α
+ (1 −α)

∫ T
t

gT−(t) −gT−(y)
(y − t)2−α

dy

)
1(0,T)(t).

The following proposition is the estimate of the generalized Stieltjes integral.

Proposition 3.1 ([7]). Fix 0 < α < 1
2

. Given two functions g ∈ W1−α,∞T (0,T)
and f ∈ Wα,1(0,T) we set

Gts(f) =

∫ t
s

frdgr.

Then for all r < t ≤ T we have

(7)

∣∣∣∣
∫ t
s

frdgr

∣∣∣∣ ≤ sup
s≤r<τ≤t

|(D1−ατ− gτ−)(r)|
∫ t
s

|(Dαs+f)(τ)|dτ

≤ Λα(g; [s,t])‖f‖α,1;[0,T]
≤ cα,T Λα(g; [s,t])‖f‖α,∞,

cα,T =

(
T +

T1−α

1 −α

)
.

4. A priori estimates

We will first deduce useful estimates for the integrals involved in Equation (2).
Let λ ∈ ( 1

2
, 1) be fixed and α ∈ (1 −λ,λ). For h ∈ Cα(0,T,IR), g ∈ Cλ(0,T,IR)

and x ∈C1−α([−r,T]), we denote

Fbt (x) :=

∫ t
0

b(s,xs)ds, G
σh
t (x) :=

∫ t
0

σh(s,xs)dhs and G
σg
t (x) :=

∫ t
0

σg(s,xs)dgs.

Proposition 4.1. Let the assumptions (Hb), (HσW ) and (HσH) be satisfied for
the coefficients b, σh and σg respectively, h ∈ Cα(0,T,IR), g ∈ Cλ(0,T,IR) and
x ∈C1−α([−r,T]). Then

(1) Fbt (x) ∈C1−α(0,T,IR).



112 TOUFIK GUENDOUZI AND SOUMIA IDRISSI

(2) G
σh
t (x) ∈C1−α(0,T,IR).

(3) G
σg
t (x) ∈C1−α(0,T,IR).

Proof.

(1) It is easy to see that Fbt (x) ∈C1(0,T,IR) and for 0 ≤ s ≤ t ≤ T

|Fbt (x) −Fbs (x)| =

∣∣∣∣∣
∫ t
0

b(u,xu)du−
∫ s
0

b(u,xu)du

∣∣∣∣∣
=

∣∣∣∣∣
∫ t
s

b(u,xu)du

∣∣∣∣∣
≤ C(1 + ‖x‖)(t−s)

≤ C(T + Tα)(1 + ‖x‖)

where C is defined as in above theorems.
Hence,

‖Fbt (x) −F
b
s (x)‖1−α ≤ C1(1 + ‖x‖1−α),

where C1 constants depending only on C,α and T .
(2) It follows from the assumption (HσW ) and the Garsia-Rodemich-Rumsey

inequality (see, Theorem 2.1.3 of [12]) that for any α ∈ (1 −λ,λ) and any
t ∈ [0,T] there exists a continuous random variable ζ(α,t,h) in t with finite
moments of any order such that

|Gσht (x) −Gσhs (x)| =

∣∣∣∣∣
∫ t
s

σh(u,xu)dhu

∣∣∣∣∣
≤ C(1 + ‖x‖)ζ(α,t,h)(t−s)

1
2
−α

≤ CT
1
2
−αζ(1 + ‖x‖),

where ζ is defined as ζ(α,t,h) = Cα

(∫ t
0

∫ t
0

|hτ −hθ|2/α

|τ −θ|1/α
dτdθ

)α/2
, Cα

constants depending only on α.
Hence,

‖Gσht (x) −G
σh
s (x)‖1−α ≤ C2ζ(1 + ‖x‖1−α),

C2 depend only on C,α and T .
(3) Let 0 ≤ s < t ≤ T . Using the proposition 3.1 we have for any α ∈ (1−λ,λ)

and λ ∈ ( 1
2
, 1)

(8)

|Gσgt (x) −G
σg
s (x)| =

∣∣∣∣∣
∫ t
s

σg(u,xu)dgu

∣∣∣∣∣
≤ Λα(g)

∫ t
s

(
|σg(θ,xθ)|
(θ −s)α

+ α

∫ θ
s

|σg(θ,xθ) −σg(v,xv)|
(θ −v)α+1

dv

)
dθ.



GLOBAL UNIQUENESS RESULT 113

Hence, by condition (HσH),

(9)

∫ t
s

|σg(θ,xθ)|
(θ −s)α

dθ ≤ CTα
(t−s)1−α

1 −α
(1 + ‖x‖1−α),

and

(10)

∫ t
s

∫ θ
s

|σg(θ,xθ) −σg(v,xv)|
(θ −v)α+1

dvdθ

≤
∫ t
s

∫ θ
s

|σg(θ,xθ) −σg(θ,xv)| + |σg(θ,xv) −σg(v,xv)|
(θ −v)α+1

dvdθ

≤
∫ t
s

∫ θ
s

C(θ −v)1−α(‖x‖1−α + (θ −v)α)
(θ −v)α+1

dvdθ

≤ C(t−s)1−α
Tα

1 −α

[
‖x‖1−α
α

+ T

]
.

Then the above estimates lead to the third assertion.

�
Consider the following equivalent norm in the space C1−α defined for any ν ≥ 0 by

‖x‖1−α,ν = sup
t
e−νt|x(t)| + sup

s<t
e−νt

|x(t) −x(s)|
(t−s)1−α

.

We now present some estimates in order to be able to use the Banach fixed point
theorem.

Proposition 4.2. Let the assumptions (Hb), (HσW ) and (HσH) be satisfied for
the coefficients b, σh and σg respectively, h ∈ Cα(0,T,IR), g ∈ Cλ(0,T,IR) and
x ∈C1−α([−r,T]). Then there exist c∗i , 1 ≤ i ≤ 3, such that

(1) ‖Fb(x)‖1−α,ν ≤ c∗1(ν)
(

1 + ‖x‖1−α,ν
)

(2) ‖Gσh(x)‖1−α,ν ≤ c∗2(ν)
(

1 + ‖x‖1−α,ν
)

(3) ‖Gσg (x)‖1−α,ν ≤ c∗3(ν)
(

1 + ‖x‖1−α,ν
)

where c∗i (ν) → 0 as ν →∞.

Proof.

(1) We remark that

e−νt(t−s)α−1|Fbt (x) −Fbs (x)| ≤ e−νt(t−s)α−1
∫ t
s

|b(u,xu)|du

≤ e−νt(t−s)α−1
∫ t
s

C(1 + ‖x‖u)du

≤ C sup
t
e−νt(t−s)α−1

∫ t
s

(1 + ‖x‖u)du

≤ C sup
t

∫ t
s

e−ν(t−u)
1 + e−νu‖xu‖

(t−u)1−α
du

≤ Cνα−1
T2α−1

2α− 1

(
1 + ‖x‖ν1−α,ν

)
,



114 TOUFIK GUENDOUZI AND SOUMIA IDRISSI

and therefore

‖Fb(x)‖1−α,ν = sup
t
e−νt|Fbt (x)| + sup

t
e−νt(t−s)α−1|Fbt (x) −F

b
s (x)|

≤
C

ν1−α(2α− 1)
(T1−α + T2α−1)

(
1 + ‖x‖ν1−α,ν

)
.

Then the above inequalities yield that there exist c∗1(ν) such that

‖Fb(x)‖1−α,ν ≤ c∗1(ν)
(

1 + ‖x‖1−α,ν
)
,

and c∗1(ν) → 0 as ν →∞.
(2) We have

|Gσht (x) −Gσhs (x)|e−νt(t−s)α−1 ≤ e−νt(t−s)α−1
∫ t
s

|σh(u,xu)|dhu

≤ e−νt(t−s)α−1CT
1
2
−αζ(α,t,h)

∫ t
s

(1 + ‖xu‖)du

≤ CT
1
2
−αζ(α,t,h) sup

t

∫ t
s

e−ν(t−u)
1 + e−νu‖xu‖

(t−u)1−α
du,

and therefore

‖Gσh(x)‖1−α,ν ≤ c∗2(ν)
(

1 + ‖x‖1−α,ν
)
,

where c∗2(ν) = C
Tα−1

(2α− 1)ν1−α
ζ(α,t,h), and c∗2(ν) → 0 as ν →∞.

(3) Using proposition 3.1 we have for s,t ∈ [0,T] with s < t
(11)

|Gσgt (x) −G
σg
s (x)|e−νt(t−s)α−1 ≤

e−νt(t−s)α−1Λα(g)
∫ t
s

(
|σg(θ,xθ)|
(θ −s)α

+ α

∫ θ
s

|σg(θ,xθ) −σg(v,xv)|
(θ −v)α+1

dv

)
dθ.

By the assumption (HσH) we have,
(12)

e−νt(t−s)α−1
∫ t
s

|σg(θ,xθ)|
(θ −s)α

dθ ≤ e−νt(t−s)α−1
∫ t
s

C
1 + ‖xθ‖
(θ −s)α

dθ

≤ CTα−1 sup
t

∫ t
s

e−ν(t−θ)
1 + e−νθ‖xθ‖

(θ −s)α
dθ

≤
C

Tα(1 − 2α)να
(

1 + ‖x‖1−α,ν
)
,

and

(13)

e−νt(t−s)α−1
∫ t
s

∫ θ
s

|σg(θ,xθ) −σg(v,xv)|
(θ −v)α+1

dvdθ

≤ e−νt(t−s)α−1
∫ t
s

∫ θ
s

C
(θ −v)1−α(‖xθ −xv‖ + 1)

(θ −v)α+1
dvdθ

≤ CTα−1 sup
t

∫ t
s

∫ θ
s

e−ν(θ−v)
1 + e−νv‖xθ −xv‖

(θ −v)2α
dvdθ

≤ C
T2α−1

(2α− 1)ν2α−1
(

1 + ‖x‖1−α,ν
)
.



GLOBAL UNIQUENESS RESULT 115

Thanks to (11)-(12) and (13) one easily remarks that there exist c∗3(ν) such
that

‖Gσg (x)‖1−α,ν ≤ c∗3(ν)
(

1 + ‖x‖1−α,ν
)
,

c∗3(ν) → 0 as ν →∞.
�

Proposition 4.3. Let the assumptions (Hb), (HσW ) and (HσH) be satisfied for
the coefficients b, σh and σg respectively, h ∈ Cα(0,T,IR), g ∈ Cλ(0,T,IR) and
x,y ∈C1−α([−r,T]). Then there exist c∗i , 1 ≤ i ≤ 3, such that

(1) ‖Fb(x) −Fb(y)‖1−α,ν ≤ c∗1(ν)||x−y||1−α,ν
(2) ‖Gσh(x) −Gσh(y)‖1−α,ν ≤ c∗2(ν)‖x−y‖1−α,ν
(3) ‖Gσg (x) −Gσg (y)‖1−α,ν ≤ c∗3(ν)(1 + ‖x‖1−α + ‖y‖1−α)‖x−y‖1−α,ν

where c∗i (ν) → 0 as ν →∞ for 1 ≤ i ≤ 3.

Proof.

(1) Using the Lipschitz property of b, we see that for 0 ≤ s < t ≤ T

e−νt(t−s)α−1
∣∣∣∣∣
(
Fbt (x)−F

b
t (y)

)
−
(
Fbs (x)−F

b
s (y)

)∣∣∣∣∣ ≤ Cνα−1(2α+1)T2α+1‖x−y‖1−α,ν
which imply the first claim.

(2) Let s,t ∈ [0,T] with s < t, we have by the Lipschitz property of σh

≤ e−νt(t−s)α−1
∣∣∣∣∣
(
G
σh
t (x) −G

σh
t (y)

)
−
(
Gσhs (x) −Gσhs (x)

)∣∣∣∣∣
≤ e−νt(t−s)α−1

∫ t
s

|σh(u,xu) −σh(u,yu)|du

≤ CT
1
2
−αζ(α,t,h) sup

t

∫ t
s

e−νu‖xu −yu‖e−ν(t−u)(t−u)α−1du.

Hence,

‖Gσh(x) −Gσh(y)‖1−α,ν ≤ C
Tα−1

α(α− 1)να−1
ζ(α,t,h)‖x−y‖1−α,ν,

which give the result of the second assertion.
(3) We have for s,t ∈ [0,T] with s < t∣∣∣(Gσgt (x) −Gσgt (y))−(Gσgs (x) −Gσgs (y))∣∣∣e−νt(t−s)α−1 ≤ e−νt(t−s)α−1Λα(g)×∫ t

s

(
|σg(θ,xθ) −σg(θ,yθ)|

(θ −s)α
+ α

∫ θ
s

|σg(θ,xθ) −σg(θ,yθ) −σg(v,xv) + σg(v,yv)|
(θ −v)α+1

dv

)
dθ.

By the Lipschitz property of σg we obtain
(14)

e−νt(t−s)α−1
∫ t
s

|σg(θ,xθ) −σg(θ,yθ)|
(θ −s)α

dθ ≤ C
T1−α

α(1 −α)να
‖x−y|‖1−α,ν.



116 TOUFIK GUENDOUZI AND SOUMIA IDRISSI

Remark that for all u,v ∈ [0,T]∣∣∣(σg(θ,xθ) −σg(θ,yθ))−(σg(v,xv) −σg(v,yv))∣∣∣
≤

∣∣∣∫ 1
0

∇σg(v,vxv + yv −vyv)
(
(xθ −yθ) − (xv −yv)

)
dv
∣∣∣

+
∣∣∣∫ 1

0

[
∇σg(θ,vxθ + yθ −vyθ) −∇σg(v,vxv + yv −vyv)

]
(xθ −yθ)dv

∣∣∣
≤ C

[
‖(xθ −yθ) − (xv −yv)‖ + ‖(xθ −yθ‖

(
‖xθ −xv‖ + ‖yθ −yv‖ + (θ −v)

)]
,

and therefore

e−νt(t−s)α−1
∫ t
s

∫ θ
s

|(σg(θ,xθ) −σg(θ,yθ)) − (σg(v,xv) −σg(v,yv))|
(θ −v)α+1

dvdθ

≤ e−νt(t−s)α−1C
∫ t
s

∫ θ
s

‖(xθ −yθ) − (xv −yv)‖
(θ −v)α+1

dvdθ

+ e−νt(t−s)α−1C
∫ t
s

∫ θ
s

‖xθ −yθ‖
(θ −v)α

dvdθ

+ e−νt(t−s)α−1C
∫ t
s

∫ θ
s

‖xθ −yθ‖(‖xθ −xv‖ + ||yθ −yv‖)
(θ −v)α+1

dvdθ

≤ (t−s)α−1C
∫ t
s

∫ θ
s

e−ν(t−θ)‖x−y‖1−α,ν
(θ −v)2α

dvdθ

+ (t−s)α−1C
∫ t
s

∫ θ
s

e−ν(t−θ)‖x−y‖1−α,ν
(θ −v)α

dvdθ

+ e−νt(t−s)α−1C
∫ t
s

∫ θ
s

‖xθ −yθ‖(‖x‖1−α + ‖y‖1−α)
(θ −v)2α

dvdθ.

Thus
(15)

e−νt(t−s)α−1
∫ t
s

∫ θ
s

|(σg(θ,xθ) −σg(θ,yθ)) − (σg(v,xv) −σg(v,yv))|
(θ −v)α+1

dvdθ

≤
CT2α−1

ν1−α(1 − 2α)
‖x−y‖1−α,ν +

CTα−1

ν(1 −α)
‖x−y‖1−α,ν

+
CT2α−1

ν1−α(1 −α)
‖x−y‖1−α,ν

(
‖x‖1−α + ‖y‖1−α

)
From (14) and (15) we can derive the estimate:

sup
s<t

∣∣∣(Gσgt (x) −Gσgt (y))−(Gσgs (x) −Gσgs (y))∣∣∣e−νt(t−s)α−1
≤ c∗3(ν)(1 + ‖x‖1−α + ‖y‖1−α)‖x−y‖1−α,ν,

c∗3(ν) → 0 as ν →∞, which implies the third claim.
�

Consider the equation on IR

ψ(x)(t) = x(0) +

∫ t
0

b(s,xs)ds +

∫ t
0

σh(s,xs)dh(s) +

∫ t
0

σg(s,xs)dg(s), t ≥ 0,

where the function ψ is defined from C1−α([−r,T]) into C1−α([−r,T]) by ψ(x)(t) =
x(t), t ∈ [−r, 0].

Proposition 4.2 and Proposition 4.3 combined together give the following result.



GLOBAL UNIQUENESS RESULT 117

Proposition 4.4. Let the assumptions (Hb), (HσW ) and (HσH) be satisfied for
the coefficients b, σh and σg respectively, h ∈ Cα(0,T,IR), g ∈ Cλ(0,T,IR) and
x,y ∈C1−α([−r,T]). Then there exist c̃i, i = 1, 2 such that

(1) ‖ψ(x)‖1−α,ν ≤‖x0‖1−α,ν + c̃1(ν)
(

1 + ‖x‖1−α,ν
)

,

(2) ‖ψ(x) −ψ(y)‖1−α,ν ≤ ‖x0 −y0‖1−α,ν + c̃1(ν)
(

1 + ‖x‖1−α + ‖y‖1−α
)
‖x−

y‖1−α,ν,
c̃i(ν) → 0, i = 1, 2, as ν →∞.

5. Deterministic functional equation

In this section we fix the parameters λ and α such that 1
2
< λ < 1, 1 −λ <

α < λ.
Consider the deterministic functional equation

(16)
x(t) = ϕ(0) +

∫ t
0

b(s,xs)ds +

∫ t
0

σh(s,xs)dh(s) +

∫ t
0

σg(s,xs)dg(s),

x0 = ϕ ∈Cr,

for g ∈Cλ(0,TIR), h ∈Cα(0,T,IR) and t ≥ 0.
The following theorem is the main result of this section.

Theorem 5.1. Let the assumptions (Hb), (HσW ) and (HσH) be satisfied for the
coefficients b, σh and σg respectively, ϕ ∈C1−α([−r, 0]). Then Eq. (16) has unique
solution x. Moreover the solution is (1 −α)-Hölder continuous on [−r,T].

Proof.

Existence. We shall prove the existence of the solution by a fixed point argu-
ment. We first define C1−α([−r,T],ϕ) as the space of all x ∈ C1−α([−r,T]) such
that x = ϕ on [−r, 0]. Let Γ be an operator defined from C1−α([−r,T],ϕ) into itself
by Γ(x)(t) = ϕ(t) for t ∈ [−r, 0] and

Γ(x)(t) = ϕ(0) +

∫ t
0

b(s,xs)ds +

∫ t
0

σh(s,xs)dh(s) +

∫ t
0

σg(s,xs)dg(s), t ≥ 0.

From Proposition 4.4. we remark that

‖Γ(x)‖1−α,ν ≤‖ϕ‖1−α,ν + c̃(ν)
(

1 + ‖x‖1−α,ν
)
,

where c̃(ν) → 0 as ν →∞.
Let ν = ν0 be sufficiently large such that c̃(ν0) ≤ 12 . If ||x||1−α,ν0 ≤ 2(1 +

||ϕ||1−α,ν0 ), then ‖Γ(x)‖1−α,ν0 ≤ 2(1 + ‖ϕ‖1−α,ν0 ) and hence Γ(Bν0 ) ⊂ Bν0 where

Bν0 =
{
x ∈C1−α([−r,T],ϕ) : ‖x‖1−α,ν0 ≤ 2(1 + ‖ϕ‖1−α,ν0 )

}
.

As consequence, Γ maps Bν0 into itself.
We now show that there exists ν > ν0 such that the operator Γ is a contraction

on Bν0 under the norm ‖ · ‖1−α,ν. Using Proposition 4.4., we have for all x,y ∈
C1−α([−r,T],ϕ)

‖Γ(x) − Γ(y)‖1−α,ν ≤ c̃(ν)(1 + ‖x‖1−α + ‖y‖1−α)‖x−y‖1−α,ν.



118 TOUFIK GUENDOUZI AND SOUMIA IDRISSI

If l0 = sup
x∈Bν0

‖x‖1−α, then for all x,y ∈ Bν0 we have

‖Γ(x) − Γ(y)‖1−α,ν ≤ c̃(ν)(1 + 2l0)‖x−y‖1−α,ν.

Let ν > ν0 be sufficiently large such that c̃(ν)(1+2l0) < 1/2. Then for all x,y ∈ Bν0
we have

‖Γ(x) − Γ(y)‖1−α,ν ≤
1

2
‖x−y‖1−α,ν.

Consequently, the operator Γ is a contraction on the closed subset Bν0 of the com-
plete metric space C1−α([−r,T]) which implies that it has a unique fixed point x
in Bν0 . So from the definition of Γ it follows that x is a solution of Eq. (16) in
C1−α([−r,T]).

Uniqueness. Assume that x,y are two solutions of (16) in the space C1−α([−r,T])
and using Proposition 4.4., with ν sufficiently large, we get

||x−y||ν ≤
1

2
||x−y||1−α,ν,

and, therefore, x = y.
�

Theorem 5.2. Let the assumptions (Hb), (HσW ) and (HσH) be satisfied for the
coefficients b, σh and σg respectively, ϕ ∈C1−α([−r, 0]). Then the solution x of Eq.
(16) satisfies

‖x‖1−α ≤ ĉ1
(

1 + ‖ϕ‖1−α
)

exp
(
ĉ2Λ

1/α
α (g)

)
,

where ĉ1, ĉ2 are constants depending only on α,T and C.

Proof. Set

J(t) = sup
s∈[−r,t]

|x(s)| + sup
−r≤s<u≤t

|x(u) −x(s)|
(u−s)1−α

, t ≥ 0.

We have for 0 ≤ s < u ≤ t,

|x(u) −x(s)|
(u−s)1−α

≤ (u−s)α−1
(∣∣∣∫ u

s

b(v,xv)dv
∣∣∣+∣∣∣∫ u

s

σh(v,xv)dh
∣∣∣+∣∣∣∫ u

s

σg(v,xv)dg
∣∣∣).

By assumption (Hb) we have

(17) (u−s)α−1
(∣∣∣∫ u

s

b(v,xv)dv
∣∣∣) ≤ C ∫ t

0

(t−v)α−1(1 + J(v))dv.

By assumption (HσW ) we get
(18)

(u−s)α−1
(∣∣∣∫ u

s

σh(v,xv)dh
∣∣∣) ≤ CT 12−αζ(α,t,h) ∫ u

s

1 + ‖xv‖
(u−v)1−α

dv

≤ CT
1
2
−αζ(α,t,h)

∫ t
0

(t−v)α−1(1 + J(v))dv.



GLOBAL UNIQUENESS RESULT 119

Using Proposition 3.1., we have for 1 −λ < α < λ and 1
2
< λ < 1

(u−s)α−1
(∣∣∣∣∣
∫ u
s

σg(v,xv)dg

∣∣∣∣∣
)

≤ (u−s)α−1Λα(g)
∫ u
s

(
|σg(v,xv)|
(v −s)α

+ α

∫ v
s

|σg(v,xv) −σg(τ,xτ )|
(v − τ)α+1

dτ

)
dv.

By the Lipschitz property of σg, we have

(19)

(u−s)α−1Λα(g)
∫ u
s

|σg(v,xv)|
(v −s)α

dv

≤ (u−s)α−1Λα(g)
∫ u
s

|σg(v,xv) −σg(v,xs)| + |σg(v,xs)|
(v −s)α

dv

≤ Λα(g)C
T2α

1 −α
+ Λα(g)C(T

1−2α + 2α)

∫ t
0

(t−s)α−1J(v)dv,

and

(20)

(u−s)α−1Λα(g)
∫ u
s

∫ v
s

|σg(v,xv) −σg(τ,xτ )|
(v − τ)α+1

dτdv

≤ (u−s)α−1Λα(g)
∫ u
s

∫ v
s

C
(v − τ)1−α(1 + ‖xv −xτ‖)

(v − τ)α+1
dτdv

≤ C
T2α−1

(2α− 1)
Λα(g)

∫ t
0

(t−v)α−1J(v)dv.

Inequalities (17), (18), (19) and (20) together imply that

sup
s<u

|x(u) −x(s)|
(u−s)1−α

≤ ĉ0(1 + Λα(g))
[
1 +

∫ t
0

(t−s)α−1J(v)dv
]
,

and

J(t) ≤‖ϕ‖1−α + ĉ0(1 + Λα(g))
[
1 +

∫ t
0

(t−s)α−1J(s)ds
]
.

Using the Gronwall lemma (see [11]), we have since

J(t) ≤‖ϕ‖1−α + ĉ0(1 + Λα(g))
[
1 +

∫ t
0

J(s)(t−s)α−1t1−αs−(1−α)ds
]

then

‖x‖1−α ≤ ĉ1(1 + ‖ϕ‖1−α) exp
(
ĉ2Λ

1/α
α (g)

)
.

�
The following result shows the dependance of the solution of Eq.(16) on the initial

condition.

Lemma 5.3. Let the assumptions (Hb), (HσW ) and (HσH) be satisfied for the
coefficients b, σh and σg respectively, ϕ,ϕ

n ∈ C1−α([−r, 0]). Let x be the solution
of Eq.(16) and xn be the solution of the same equation with ϕn in place of ϕ. Then
for 1 −λ < α < λ and 1

2
< λ < 1 we have

‖x−xn‖1−α ≤ ĉ1‖ϕ−ϕn‖1−α exp
(
ĉ2(‖x‖

1/α
1−α + ‖x

n‖1/α1−α)
)

exp
(
ĉ3Λ

1/α
α (g)

)
,

where ĉ1, ĉ2 and ĉ3 depend only on α,T and C.



120 TOUFIK GUENDOUZI AND SOUMIA IDRISSI

Proof. For t ≤ 0, let

Jn(t) = sup
−r≤s≤t

|x(s) −xn(s)| + sup
s<u

|x(u) −xn(u) −x(s) + xn(s)|
(u−s)1−α

.

Following the same lines as in Proof of Theorem 5.2. we obtain for t ≤ 0

Jn(t) ≤‖ϕ−ϕn‖1−α + ĉ0
[
1 + Λα(g)(1 +‖x‖1−α +‖xn‖1−α)

]∫ t
0

Jn(s)(t−s)α−1ds.

Therefore,

Jn(t) ≤‖ϕ−ϕn‖1−α+ĉ0
[
1+Λα(g)(1+‖x‖1−α+‖xn‖1−α)

]∫ t
0

J(s)(t−s)α−1t1−αs−(1−α)ds.

By the Gronwall lemma ([11]) we get

‖x−xn‖1−α ≤ ĉ1‖ϕ−ϕn‖1−α exp
(
ĉ3Λ

1/α
α (g)

)
exp

(
ĉ2(‖x‖

1/α
1−α + ‖x

n‖1/α1−α)
)
.

�

6. Functional equation driven by a Wiener process and fBm

In this section we apply the deterministic results in order to prove the main
theorems of this article.

Proof.(Theorem 2.1)
The existence and uniqueness of the solution can be established following the

same argument as in the deterministic Theorem 5.1.
Using Theorem 5.2., we get for α > 1 −H

‖x‖1−α ≤ ĉ1(1 + ‖φ‖1−α) exp
(
ĉ2Λ

1/α
α (B)

)
,

ĉ1, ĉ1 depend only on α,T and C.
Therefore, for all p ≥ 1 we have

(21) IE‖x‖p1−α ≤
1

2
ĉ
2p
1 IE(1 + ‖φ‖1−α)

2p +
1

2
IE exp

(
2pĉ2Λ

1/α
α (B)

)
.

Hence for any 0 < γ < 2 we have by Fernique’s theorem ([3])

IE
[

exp Λα(B)
γ
]
< ∞.

As consequence IE||x||p1−α < ∞, ∀p ≥ 1 such that
1

α
< 2 with H should be greater

than 3
4

and α + H > 3
2
.

�

Proof.(Theorem 2.2)
The almost-sure convergence can be obtained using Lemma 5.3. The ILp-convergence

can also be obtained by a dominated convergence argument since we have for any
n ≥ 0

‖xn −x‖1−α ≤ ‖xn‖1−α + ‖x‖1−α
≤ ĉ1(2 + ‖φ‖1−α + ‖φn‖1−α) exp

(
ĉ2Λ

1/α
α (B)

)
and ||φn||1−α is bounded, the we can write

|‖xn −x‖≤ ĉ4 exp
(
ĉ2Λ

1/α
α (B)

)
:= Y



GLOBAL UNIQUENESS RESULT 121

and IEY p < ∞, ∀p ≥ 1.
�

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Laboratory of Stochastic Models, Statistic and Applications, Tahar Moulay Uni-

versity PO.Box 138 En-Nasr, 20000 Saida, Algeria

∗Corresponding author