Tan Advances in Difference Equations 2014, 2014:321
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RESEARCH
Open Access
Exponential stability of fractional stochastic
differential equations with distributed delay
Li Tan*
*
Correspondence: [email protected]
School of Statistics, Jiangxi
University of Finance and
Economics, Nanchang, Jiangxi
330013, China
Abstract
Equations driven by fractional Brownian motion are attracting more and more
attention. This paper considers fractional stochastic differential equations with
distributed delay. With the variation-of-constants formula, an explicit expression and
asymptotic behavior of the solution are provided, sufficient conditions are derived to
guarantee the pth moment exponential stability and almost surely exponential
stability.
MSC: 39B82; 60H05
Keywords: fractional Brownian motion; variation-of-constants formula; exponential
stability; distributed delay
1 Introduction
Fractional Brownian motion (fBm) was first studied in  by Kolmogorov [], who defined it in a Hilbert space and named it a Wiener helix. Until , Mandelbrot and Van
Ness [] provided the stochastic integral representation of fBm in terms of a standard
Brownian motion, the name fBm was then introduced. As an extension of classic Brownian motion, fBm can give a better description to model natural situations like the temperature at a specific place as a function of time and so on []. Because of their extensive
applications in finance, economics, biology, etc., fBm is attracting more and more attention.
An fBm with Hurst index H ∈ (, ) is a continuous and centered Gaussian process
{BH (t), t ≥ } with covariance function

E BH (t)BH (s) = t H + sH – |t – s|H .

For H =  , the fBm is exactly the standard Brownian motion without memory. For H =  ,
it is usually divided into two cases,  < H <  and  < H < . The case for H >  is called a
long memory process, while H <  is called a short memory process.
It is known that fBm shares something with Brownian motion, but the critical difference
is that the increments of fBm are dependent while that of Brownian motion are independent. Therefore, fBm is neither a Markov process, nor a semimartingale, and the theories
of stochastic differential equations (SDEs) driven by standard Brownian motion are invalid in the case of fBm. Recently, there are some studies on existence, uniqueness, and
stability of the solutions for fractional SDEs. Ferrante and Rovira [] proved the existence
©2014 Tan; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any
medium, provided the original work is properly cited.
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and uniqueness for a stochastic delay differential equation driven by a fractional Brownian motion with Hurst parameter H > /. Ferrante and Rovira [] gave a proof of the
existence and uniqueness for stochastic delay differential equations with hereditary drift
driven by a fractional Brownian motion with Hurst parameter H > /. Caraballo et al. []
investigated the existence, uniqueness, and exponential asymptotic behavior of mild solutions to stochastic delay evolution equations perturbed by a fractional Brownian motion
with Hurst parameter H > /. Based on the research of [], Boufoussi and Hajji [] gave a
proof of the existence and uniqueness of mild solutions for a neutral stochastic differential equation with finite delay, driven by a fractional Brownian motion in a Hilbert space.
Dung [] gave a sufficient condition for the exponential asymptotic behavior of solutions
of a general class of linear fractional SDEs with time-varying delays. However, to the best
of our knowledge, fractional SDEs with distributed delay have not been considered till
now. In this paper, SDEs driven by an fBm with distributed delay will be introduced and
asymptotic behavior of the solution will be analyzed.
In [], Mandelbrot and Van Ness discussed the following integral representation:
t

H–/
B (t) = B () +
Zt + (t – s)
dW (s) ,
(H + /)

H
H
t > ,
where

Zt =
(t – s)H–/ – (–s)H–/ dW (s),
–∞
W (t) is a standard Brownian motion, represents the gamma function, and  < H <  is
the Hurst parameter. There are also several other stochastic integral representations for
fBm [].
As to the presentation introduced by Mandelbrot and Van Ness [], the integral form
t
W H (t) =
(t – s)H–/ dW (s)

is called the Riemann-Liouville fractional integral. There is a relation between the
Riemann-Liouville fractional Brownian motion (RLfBm) and fBm []. Considering that
the process Zt has absolutely continuous trajectories, it suffices to consider the term W H (t)
instead of BH (t) [], thus, BH (t) will be denoted by W H (t) along our paper.
This paper is organized as follows: In Section , the explicit form of solution to the
retarded SDE with an fBm is given. In Section , asymptotic behavior of the solution is
provided, and sufficient conditions are derived to guarantee the pth moment exponential
stability and almost surely exponential stability.
2 Preliminary and explicit solution
In the following sections, we consider retarded SDE driven by an fBm in the form:

dX(t) =
X(t + θ )ρ(dθ ) dt + σ (t) dW H (t),
t>
(.)
–τ
with the initial value X(t) = ξ (t), t ∈ [–τ , ], where W H (t) is a RLfBm with Hurst parameter
H >  , ρ(·) is a finite signed measure defined on [–τ , ].
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In order to give the explicit form of (.), we first consider the following deterministic
retarded differential equation:

dY (t) =
Y (t + θ )ρ(dθ ) dt
(.)
–τ
with the initial value Y (t) = ξ (t), t ∈ [–τ , ]. It is easy to see that the characteristic equation
of (.) is

eλθ ρ(dθ ) = .
h(λ) := λ –
–τ
Denote Z(t) as the fundamental solution of (.) with initial value Z() =  and Z(θ) = ,
θ ∈ [–τ , ). By the variation-of-constants formula (see, e.g., []), the solution of (.) has
a unique explicit form:
 
Y (t) = Z(t)ξ () +
Z(t + θ – s)ξ (s) dsρ(dθ ),
–τ
t ≥ .
θ
According to Hale [], for any α > α := max{Re(λ) : h(λ) = }, there exists kα >  such
that the fundamental solution Z(t) satisfies the inequality
Z(t) ≤ kα eαt ,
t ≥ –τ .
Before we introduce the explicit representation of (.), we first present a lemma that is
useful in later parts.
Lemma . [] For every > , denote by β = H –  , define
W
H,
t
(t – s + )β dW (s).
(t) =

Then {W H,
(t)}t≥ is a semimartingale with the following decomposition:
W
H,
t
β
φ (s) ds,
(t) = W (t) +
(.)

s
where φ (s) =  β(s + – u)β– dW (u). Furthermore, W H,
(t) converges to W H (t) in L ()
uniformly with respect to t ∈ [, T] when → .
Theorem . There is a unique strong solution {X(t)}t≥ to (.) which reads
 
Z(t + θ – s)ξ (s) dsρ(dθ )
X(t) = Z(t)ξ () +
–τ
θ
t
Z(t – s)σ (s) dW H (s),
+
t ≥ ,
(.)

where Z(t) is the fundamental solution of (.) with initial value Z() =  and Z(θ) = ,
θ ∈ [–τ , ).
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Proof According to Dung [], the solution X(t) of (.) can be approximated by solving
the following equation:

dX (t) =
X (t + θ )ρ(dθ ) dt + σ (t) dW H,
(t),
t>
–τ
with the initial datum X (t) = ξ (t), t ∈ [–τ , ]. By the decomposition (.), we can rewrite
the above equation as

dX (t) =
X (t + θ )ρ(dθ ) + σ (t)φ (t) dt + σ (t)
β dW (t),
t > .
–τ
Multiplying this equation by e–λt , Re λ > c sufficiently large, integrating from  to ∞, denoting by L (X )(λ) the Laplace transform of X (t), we obtain
h(λ)L X (λ) = ξ () +
eλ(θ –t) ξ (t) dtρ(dθ ) +
–τ
∞
+
 
∞
e–λt σ (t)φ (t) dt

θ
e–λt σ (t)
β dW (t).

An application of the Laplace inversion theorem (see, for example, []) yields
e h (λ) ξ () +
 
λt –
X (t) =
eλ(θ –t) ξ (t) dtρ(dθ )
–τ
(c)
∞
+
–λt
e
θ
σ (t)φ (t) dt +

∞
–λt
e
β
σ (t)
dW (t) dλ.

Note that
L Z
(λ) = h– (λ),
where Z (t) is the fundamental solution of

dY (t) =
Y (t + θ )ρ(dθ ) dt
–τ
with the initial value Z () =  and Z (θ) = , θ ∈ [–τ , ).
With the definition of a convolution of the Laplace transform, we get
 
X (t) = Z (t)ξ () +
Z
(t + θ – s)ξ (s) dsρ(dθ )
–τ
θ
t
Z
(t – s)σ (s)φ (s) ds +
+

t
Z
(t – s)σ (s)
β dW (s).

Therefore, by (.), it follows that
 
X (t) = Z (t)ξ () +
Z
(t + θ – s)ξ (s) dsρ(dθ )
–τ
θ
t
Z
(t – s)σ (s) dW H,
(s),
+
t ≥ .

Let → , the representation of X(t) is obvious.
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3 Exponential behavior
The following two theorems provide the pth moment exponential stability and almost sure
exponential stability of the solutions to (.).
Theorem . Assume that α < , and there exists γ >  such that
∞

eγ s σ (s) ds < ∞,
(.)

then there exist positive constants K and μ such that for each p > , and the solution of
(.) satisfies
p
EX(t) ≤ Kξ p∞ e–μpt ,
t ≥ ,
where ζ ∞ := sup–τ ≤θ ≤ |ζ (θ)|.
Proof According to the explicit form (.), along with the inequality |a + b + c|p ≤ kp (|a|p +
|b|p + |c|p ), p, kp > , it is easy to get
 
p
Z(t + θ – s)ξ (s) dsρ(dθ )
E X(t) = EZ(t)ξ () +
–τ θ
p
t
+
Z(t – s)σ (s) dW H (s)

p ≤ kp E Z(t)ξ () +  
–τ
θ
p
Z(t + θ – s)ξ (s) dsρ(dθ )
t
p H
+ Z(t – s)σ (s) dW (s)

t
p
–αpt
p
H
≤ K e
ξ ∞ + kp E Z(t – s)σ (s) dW (s) .
(.)

Denote by
t
p
I(t) := kp E Z(t – s)σ (s) dW H (s) .

We should remind the reader that fBm is a centered Gaussian process. So, before consid
t
ering I(t), we can give an estimation on |  Z(t – s)σ (s) dW H (s)| first. Applying (.), we
obtain
t
Z(t – s)σ (s) dW H (s)

t
= lim
→ 
Z(t – s)σ (s) dW H,
(s)
t
s
Z(t – s)σ (s)(s – u)β– dW (u) ds
=β

t

t
Z(t – s)σ (s)(s – u)β– ds dW (u).
=β

u
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By the Itô isometry formula and the Hölder inequality, we have
t

E Z(t – s)σ (s) dW H (s)

t 
t
= β
Z(t – s)σ (s)(s – u)β– ds

t ≤ β

≤ tβ
du
u
Z(t – s)σ (s) (s – u)β– ds
t
u
Z(t – s)σ (s) sβ ds ≤ t β
t

≤ kα t β
t
(s – u)β– ds du
u
Z(t – s)σ (s) ds
t

t

e–α(t–s) σ (s) ds.

By virtue of [, Lemma .] and condition (.), letting γ = α ∧ γ , we have
t

E Z(t – s)σ (s) dW H (s) ≤ kα t β e–γ t

∞


eγ s σ (s) ds ≤ K e–γ t ,
∞
where K = kα t β e–γ t  eγ s |σ (s)| ds, γ =  γ . Therefore, the Gaussian property leads
p
to I(t) ≤ K e–γ pt . An application of I(t) on (.) yields
p
EX(t) ≤ Kξ p∞ e–μpt ,
t ≥ ,
where μ = α ∧ γ . This completes the proof.
Theorem . Assume that (.) is satisfied, then there exists a positive constant ν such
that the solution of (.) has the property

logX(t) ≤ –ν.
t→∞ t
lim sup
Proof For n –  ≤ t ≤ n, n ≥ , X(t) can be represented as
t

X(t) = X(n – ) +
X(s + θ )ρ(dθ ) ds +
–τ
n–
t
σ (s) dW H (s).
n–
With the inequality |a + b + c| ≤ (|a| + |b| + |c| ), we derive from (.) that
E

 sup X(t) ≤ EX(n – ) + E
n–≤t≤n
+ E sup n–≤t≤n
≤
where J(t) = E(supn–≤t≤n |
t
n– σ (s) dW
H
n
n–
t


X(s + θ )ρ(dθ ) ds
–τ
 σ (s) dW (s)
H
n–
C ξ ∞ e–μ(n–)
+ C e–μ(n–) + J(t),
(s)| ).
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Next, by the Burkhold-Davis-Gundy inequality, (.) and (.), we arrive at
sup n–≤t≤n
J(t) ≤ E
≤ C
n
t
n–
 σ (s) dW H (s)
n
σ (s)β(s – u)β– ds
n–

du ≤ C e–γ (n–) .
u
Let γ = μ ∧ γ , we see that
E
 sup X(t) ≤ Ce–γ n .
n–≤t≤n
Then, with the Chebyshev inequality, for any γ < γ ,

 P sup X(t) > e–γ n ≤ eγ n E sup X(t) ≤ Ce–(γ –γ )n .
n–≤t≤n
n–≤t≤n
–(γ –γ )n
Since ∞
< ∞, by virtue of Borel-Cantelli lemma, there exists an  ∈ with
n= e
P( ) =  such that for every ω ∈  , there exists an integer n (ω), for n ≥ n (ω), n –  ≤
t ≤ n,
X(t) ≤ e–γ n ≤ e–γ t .
The desired conclusion is satisfied with ν = γ .
Remark . This provides a method to ensure the exponential stability of SDEs driven by
an fBm, and it can be generalized to fractional SDEs of neutral type.
Competing interests
The author declares to have no competing interests.
Author’s contributions
The author carried out the proof and conceived of the study. The author read and approved the final manuscript.
Acknowledgements
The author is very grateful to the editor and the anonymous referees for their insightful and constructive comments and
suggestions, which have led to an improved version of this paper. The author also thanks the NNSF (71371193) for
support.
Received: 20 September 2014 Accepted: 4 December 2014 Published: 18 Dec 2014
References
1. Kolmogorov, AN: Wienersche Spiralen und einige andere interessante Kurven im Hilbertschen Raum. C. R. (Dokl.)
Acad. Sci. URSS 26, 115-118 (1940)
2. Mandelbrot, BB, Van Ness, JW: Fractional Brownian motions, fractional noises and applications. SIAM Rev. 10, 422-437
(1968)
3. Biagini, F, Hu, YZ, Øksendal, B, Zhang, TS: Stochastic Calculus for Fractional Brownian Motion and Applications.
Springer, London (2008)
4. Ferrante, M, Rovira, C: Stochastic delay differential equations driven by fractional Brownian motion with Hurst
parameter H > 1/2. Bernoulli 12, 85-100 (2006)
5. Ferrante, M, Rovira, C: Convergence of delay differential equations driven by fractional Brownian motion. J. Evol. Equ.
10, 761-783 (2010)
6. Caraballo, T, Garrido-Atienza, MJ, Taniguchi, T: The existence and exponential behavior of solutions to stochastic
delay evolution equations with a fractional Brownian motion. Nonlinear Anal. 74, 3671-3684 (2011)
7. Boufoussi, B, Hajji, S: Neutral stochastic functional differential equations driven by a fractional Brownian motion in a
Hilbert space. Stat. Probab. Lett. 82, 1549-1558 (2012)
Tan Advances in Difference Equations 2014, 2014:321
http://www.advancesindifferenceequations.com/content/2014/1/321
8. Dung, N: Asymptotic behavior of linear fractional stochastic differential equations with time-varying delays.
Commun. Nonlinear Sci. Numer. Simul. 19, 1-7 (2014)
9. Lim, SC, Sithi, VM: Asymptotic properties of the fractional Brownian motion of Riemann-Liouville type. Phys. Lett. A
206, 311-317 (1995)
10. Alòs, E, Mazet, O, Nualart, D: Stochastic calculus with respect to fractional Brownian motion with Hurst parameter
lesser than 1/2. Stoch. Process. Appl. 6, 121-139 (2000)
11. Bao, JH, Yin, G, Yuan, CG: Stationary distributions for retarded stochastic differential equations without dissipativity.
arXiv:1308.2018
12. Hale, JK: Theory of Functional Differential Equations. Springer, Berlin (1977)
13. Thao, TH: An approximate approach to fractional analysis for finance. Nonlinear Anal. 7, 124-132 (2006)
10.1186/1687-1847-2014-321
Cite this article as: Tan: Exponential stability of fractional stochastic differential equations with distributed delay.
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