R E S E A R C H
Open Access
The averaging method for stochastic
differential delay equations under
non-Lipschitz conditions
Li Tan
*and Dingyou Lei
*Correspondence:
School of Traffic and Transportation Engineering, School of
Mathematics, Central South University, Changsha, Hunan 410075, P.R. China
Abstract
Under a non-Lipschitz condition, the averaging principle for the general stochastic differential delay equations (SDDEs) is established. The solutions of convergence in mean square and convergence in probability between standard SDDEs and the corresponding averaged SDDEs are considered.
MSC: 34K50; 34C29
Keywords: averaging method; stochastic differential delay equations; non-Lipschitz
1 Introduction
The averaging principle plays an important role in dynamical systems in problems of me-chanics, physics, control and many other areas. The rigorous results on averaging princi-ples were firstly put forward by Krylov and Bogolyubov []. After that, Khasminskii [, ] considered averaging principles of Itô’s stochastic differential equations, parabolic and el-liptic differential equations.
Most systems in science and industry are perturbed by some random environmental ef-fects, which are often described as Gaussian noise. As a special non-Gaussian Lévy noise, Poisson noise is usually a hot spot when dealing with stochastic systems. Stoyanov and Bainov [] investigated the averaging method for a class of stochastic differential equa-tions with Poisson noise. They considered the connecequa-tions between the soluequa-tions of a standard form and the solutions of averaged systems and proved that under some con-ditions the solutions of averaged systems converge to the solutions of original systems in mean square and in probability. Following the Bogolyubov theorem (cf.[]), Kolomiets and Mel’nikov [] gave a theorem concerning averaging on the finite time interval of a sys-tem of integral-differential equations with Poisson noise. Instead of Poisson noise, Xuet al.[] established an averaging principle for stochastic differential equations with general non-Gaussian Lévy noise.
Stochastic differential equations (SDEs) give models for systems, the future state of which is independent of the past states and is determined solely by the present. How-ever, for real systems, the future state will often be related to past states except for the present states of the system. Thus, SDDEs have attracted great attention recently [, ], but averaging principles for SDDEs have not been considered so far. Motivated by the previous paper, we consider the averaging principles for general SDEs and SDDEs with a non-Lipschitz condition.
The following sections are organized as follows. In Section we give a detailed descrip-tion on the averaging process of general SDEs under a non-Lipschitz condidescrip-tion. Secdescrip-tion shows the averaging principles for SDDEs under a non-Lipschitz condition.
2 Stochastic differential equation case
Let (,F,P) be a complete probability space with filtration{Ft}t≥satisfying the usual
conditions (i.e., it is right continuous andFcontains allPnull sets), and letB(t) be a
givenm-dimensional Brownian motion defined on the space. Let <T<∞andXbe an
F-measurableRd-valued random variable such thatE|X|<∞. Consider the following d-dimensional stochastic differential equation:
dX(t) =ft,X(t)dt+gt,X(t)dB(t), ≤t≤T, ()
with initial data X, wheref : [,T]×Rd→Rd andg: [,T]×Rd→Rd×m are both continuous and Borel measurable. By the definition of stochastic differential, this equation is equivalent to the following stochastic integral equation:
X(t) =X+
t
fs,X(s)ds+
t
gs,X(s)dB(s), t∈[,T]. ()
In order to guarantee the existence and uniqueness of the solution to (), we impose a condition on the coefficient functions.
(A) Non-Lipschitz condition: for anyx,y∈Rdandt∈[,T],
f(t,x) –f(t,y)∨g(t,x) –g(t,y)≤κ|x–y|,
whereκ(·) is a continuous increasing concave function fromR+toR+such thatκ() = ,
κ(x) > forx> and
+
dx
κ(x)=∞.
It is known from [, Theorem .] that under the condition (A), there exists a unique solutionX(t) to () with the initial dataX.
The standard form of () is
X(t) =X+
t
fs,X(s)
ds+√
t
gs,X(s)
dB(s), ()
whereXand the coefficients have the same conditions as in (), and∈(,] is a positive
small parameter witha fixed number.
According to the existence and uniqueness theorem of differential equations, () also has a unique solutionX(t),t∈[,T] for every fixed∈(,]. In order to find out whether
the solutionX(t) will be approximated with small to some other simpler process, we
impose some conditions on the coefficients.
Forx∈RdandT∈[,T], (A)
T
T
f(s,x) –f¯(x)ds≤ψ(T)
+|x|,
(A)
T
T
g(s,x) –g¯(x)ds≤ψ(T)
+|x|,
whereψi(T),i= , are positive bounded functions withlimT→∞ψi(T) = .
We now consider the following averaged stochastic equation which corresponds to the original standard form ():
Y(t) =X+
t
¯
fY(s)
ds+√
t
¯ gY(s)
dB(s). ()
Obviously, () also has a unique solutionY(t) under similar conditions as () for the
solutionX(t). Now, we consider the connections between the processesX(t) andY(t).
The convergence in mean square and convergence in probability between the standard form and the averaged form of () are especially considered.
The following two theorems give the connections between the processesX(t) andY(t).
Theorem . Suppose that the conditions(A)-(A)are satisfied.For a given arbitrarily small numberδ> and a constant L> ,α∈(, ),there exists a number∈(,]such that for all∈(,],we have
E sup t∈[,L–α]
X(t) –Y(t)
≤δ.
Proof Consider the differenceX(t) –Y(t). By () and (), we have
X(t) –Y(t) =
t
fs,X(s)
–f¯Y(s) ds+
√
t
gs,X(s)
–g¯Y(s) dB(s).
By the elementary inequality|x+x|≤(|x|+|x|), foru∈[,T], we obtain
sup
≤t≤u
X(t) –Y(t)
≤ sup
≤t≤u
tfs,X(s)
–¯fY(s) ds
+ sup
≤t≤u
tgs,X(s)
–g¯Y(s) dB(s)
.
Denote by
J= sup
≤t≤u
tfs,X(s)
–f¯Y(s) ds
,
J= sup
≤t≤u
tgs,X(s)
–g¯Y(s) dB(s)
Thus, using the elementary inequality again, we get
By the Cauchy-Schwarz inequality and the condition (A), taking expectation on J
By the properties of solutions to stochastic differential equations, we know that if
E|X|<∞, then for eacht≥,E|X(t)|<∞(cf.[]). Following the discussion of [], this property combines with the fact thatlimT→∞ψ(T) = . We can further estimate
that there exists a constantCsuch that
E|J|≤uC.
On the other hand, forJ, taking expectation on it, using the Burkholder-Davis-Gundy inequality [] and the elementary inequality again, we get
+ E
u
gs,Y(s)
–g¯Y(s)
ds
:=E|J|+E|J|.
By the condition (A), we obtain
E|J|≤
u
E
κX(s) –Y(s)
ds.
As toE|J|, we use the condition (A) to get
E|J|≤E
u
gs,Y(s)
–g¯Y(s)
ds
≤uψ(u)
+Esup
≤t≤u
Y(t)
.
As a similar way of dealing withE|J|, there exists a constantCsuch that
E|J|≤uC.
Then
E|J|≤
u
EκX(s) –Y(s)
ds+ uC. ()
Putting () and () together, we see that
Esup
≤t≤u
X(t) –Y(t)
≤(u+ )
u
EκX(s) –Y(s)
ds+ u(uC+ C). ()
From the condition (A), we know thatκis concave andκ() = , therefore, we can find a pair of positive constantsaandbsuch that
κ(x)≤a+bx for allx≥. ()
Substituting this into () gives that
Esup
≤t≤u
X(t) –Y(t)
≤b(u+ )
u
EX(s) –Y(s)
ds+ u(ua+ uC+ a+ C)
≤u(ua+ uC+ a+ C) + b(u+ )
u
E sup
≤s≤s
X(s) –Y(s)
ds.
The Gronwall inequality then yields
Esup
≤t≤u
X(t) –Y(t)
Chooseα∈(, ) andL> such that for everyt∈[,L–α]⊆[,T],
E sup
≤t≤L–α
X(t) –Y(t)
≤CL–α,
whereC= (aL–α+ CL–α+ a+ C)exp{bL–α(L–α+ )}is a constant.
That is, given any numberδ> , we can choose∈(,] such that for each∈(,]
and for everyt∈[,L–α],
E sup t∈[,L–α]
X(t) –Y(t)
≤δ.
This completes the proof.
Except for the convergence in mean square between () and (), we can also get the properties of convergence in probability.
Theorem . Suppose that the conditions(A)-(A)are satisfied,for a given arbitrarily small numberδ> and a constant L> ,α∈(, ),there exists a number∈(,]such that for all∈(,],we have
lim
→P
sup t∈[,L–α]
X(t) –Y(t)>δ
= .
Proof With the result of Theorem . and the Chebyshev-Markov inequality, for any given numberδ> , we have
P sup t∈[,L–α]
X(t) –Y(t)>δ
≤ δE
sup t∈[,L–α]
X(t) –Y(t)
≤ δCL
–α
.
Taking limits on both sides of the inequality, we get the required result.
In order to illustrate the averaging process of stochastic differential equations under non-Lipschitz conditions, we give the following example.
Example . Consider the one dimensional SDE
dX(t) =sintX(t) dt+X(t) dB(t), t≥,
with an initial conditionX() =X, andE|X|<∞, whereB(t) is a scalar Brownian
mo-tion. The corresponding standard form of the above SDE is
dX=sintXdt+
√
XdB(t).
Denote
f(t,X) =sintX, g(t,X) =X.
Then
¯
f(X) =
π
π
f(t,X) dt=
πX, g¯(X) = π
π
Define the averaged SDE
dY=
πYdt+ √
YdB(t).
Obviously, the explicit solution of this equation is
Y(t) =Xexp
π –
t+B(t)
.
Forκ(x) =kx, wherekis a positive constant, it is easy to see that the conditions (A)-(A) are satisfied, thus Theorems . and . hold. That is,
E sup t∈[,L–α]
X(t) –Y(t)
≤δ
andX(t)→Y(t),→ in probability.
As a similar process to the stochastic differential equation case, we can derive the aver-aging principle for SDE with delay.
3 Stochastic differential delay equation case
Letτ> and denote byC([–τ, ];Rd) the family of continuous functionsφfrom [–τ, ] to
Rdwith the normφ=sup
–τ≤θ≤|φ(θ)|. Let <T<∞. We now consider the following
stochastic differential delay equation inRd:
dX(t) =ft,X(t),X(t–τ)dt+gt,X(t),X(t–τ)dB(t), ≤t≤T, ()
wheref : [,T]×Rd×Rd→Rdandg: [,T]×Rd×Rd→Rd×mare both continuous and Borel measurable. Suppose the initial dataX() =ξ={ξ(θ) : –τ≤θ≤}, which is an
F-measurableC([–τ, ];Rd)-valued random variable such thatEξ<∞.
We impose the following condition:
(A) Non-Lipschitz condition: for anyx,xˆ,y,yˆ∈Rdandt∈[,T],
f(t,x,y) –f(t,xˆ,yˆ)∨g(t,x,y) –g(t,xˆ,yˆ)≤κ|x–xˆ|+K|y–ˆy|, ()
whereKis a positive constant andκ(·) is a continuous increasing concave function from
R+toR+such thatκ() = ,κ(x) > forx> and
+
dx
κ(x)=∞.
Under the condition (A), () has a unique solution fort∈[,T]. Consider the standard form of SDDE inRd:
X(t) =X() +
t
fs,X(s),X(s–τ)
ds+√
t
gs,X(s),X(s–τ)
dB(s), ()
where X() and the coefficients have the same conditions as in (), and∈(,] is a
Definef¯(x,y) :Rd×Rd→Rd,g¯(x,y) :Rd×Rd→Rd×m to be measurable functions, satisfying the condition (A). Moreover, we assume that the following inequalities are satisfied:
Forx,y∈RdandT∈[,T], (A)
T
T
f(s,x,y) –f¯(x,y)ds≤ψ(T)
+|x|+|y|, ()
(A)
T
T
g(s,x,y) –g¯(x,y)ds≤ψ(T)
+|x|+|y|, ()
whereψi(T),i= , are positive bounded functions withlimT→∞ψi(T) = . The averaged form of () is
Z(t) =X() +
t
¯
fZ(s),Z(s–τ)
ds+√
t
¯
gZ(s),Z(s–τ)
dB(s). ()
Obviously, () also has a unique solutionZ(t) under similar conditions as () for the
solutionX(t). In the rest of the paper, we will consider the connections between the
pro-cessesX(t) andZ(t).
Theorem . Suppose that the conditions(A)-(A)are satisfied.For a given arbitrarily small numberδ> and a constant L> ,α∈(, ),there exists a number∈(,]such that for all∈(,],we have
E sup t∈[,L–α]
X(t) –Z(t)
≤δ.
Proof Considering the differenceX(t) –Z(t), we have
X(t) –Z(t) =
t
fs,X(s),X(s–τ)
–f¯Z(s),Z(s–τ) ds
+√
t
gs,X(s),X(s–τ)
–g¯Z(s),Z(s–τ) dB(s).
Foru∈[,T], it is easy to obtain by elementary inequalities that
sup
≤t≤u
X(t) –Z(t)
≤ sup
≤t≤u
tfs,X(s),X(s–τ)
–f¯Z(s),Z(s–τ) ds
+ sup
≤t≤u
t
gs,X(s),X(s–τ)
–g¯Z(s),Z(s–τ) dB(s)
By elementary computation, we get
By the Cauchy-Schwarz inequality and the condition (A), taking expectation onK
Taking expectation onK
and using the Burkholder-Davis-Gundy inequality, we get
With the condition (A),
By (), we can further derive
Esup
The Gronwall inequality then yields
Esup
Chooseα∈(, ) andL> such that for everyt∈[,L–α]⊆[,T],
E sup
≤t≤L–α
X(t) –Z(t)
≤CL–α,
whereC= [C+ a(L–α+ )]exp{L–α(L–α+ )(b+K)}is a constant.
That is, given any numberδ> , we can choose∈(,] such that for each∈(,]
and for everyt∈[,L–α],
E sup t∈[,L–α]
X(t) –Z(t)
≤δ.
This completes the proof.
The next theorem gives convergence in probability betweenX(t) andZ(t).
Theorem . Suppose that the conditions(A)-(A)are satisfied.For a given arbitrarily small numberδ> and a constant L> ,α∈(, ),there exists a number∈(,]such that for all∈(,],we have
lim
→P
sup t∈[,L–α]
X(t) –Z(t)>δ
= .
Proof With the result of Theorem . and the Chebyshev-Markov inequality, for any given numberδ> , we have
P sup t∈[,L–α]
X(t) –Z(t)>δ
≤ δE
sup t∈[,L–α]
X(t) –Z(t)
≤ δCL
–α
.
Taking limits on both sides of the inequality, we get the required result.
The following example gives the averaging process of stochastic differential delay equa-tions under non-Lipschitz condiequa-tions.
Example . Consider the following one-dimensional SDDE:
dX(t) = sint
aX(t) +bX(t– )
dt+√cX(t) +dX(t– )
dB(t), t≥,
with an initial conditionX(t) =t+ ,t∈[–, ], wherea,b,c,dare constants andB(t) is a
one-dimensional Winner process. Obviously,f(t,x,y) = sint(ax+by),g(t,x,y) =cx+dy. Let
¯
fX(t),X(t– )
= π
π
ft,X(t),X(t– )
dt=aX(t) +bX(t– ),
¯
gX(t),X(t– )
= π
π
gt,X(t),X(t– )
dt=cX(t) +dX(t– ).
Define the corresponding averaged SDDE as follows:
dZ(t) =
aZ(t) +bZ(t– )
dt+√cZ(t) +dZ(t– )
Ont∈[, ], the linear SDDE becomes a linear SDE
dZ(t) =aZ(t) +btdt+cZ(t) +dtdB(t).
The explicit solution of this SDE is
Z(t) =(t)
X() +
t
(b–cd)–(s)sds+
t
d–(s)sdB(s)
,
where
(t) =exp
a– c
t+cB(t)
.
Repeating this procedure over the intervals [, ], [, ],etc.we can obtain the explicit solution.
Forκ(x) =kx, wherekis a positive constant, it is easy to see that the conditions (A )-(A) are satisfied, thus Theorems . and . hold. That is,
E sup t∈[,L–α]
X(t) –Z(t)
≤δ,
andX(t)→Z(t),→ in probability.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors carried out the proof and conceived of the study. All authors read and approved the final manuscript.
Acknowledgements
The authors are very grateful to the editor and the anonymous referees for their insightful and constructive comments and suggestions that have led to an improved version of this paper. The author would like to thank Central South University for the Postdoctoral Funds and the Fundamental Research Funds.
Received: 5 December 2012 Accepted: 31 January 2013 Published: 19 February 2013 References
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doi:10.1186/1687-1847-2013-38
