On the averaging principle for stochastic delay differential equations with jumps

In this paper, we investigate the averaging principle for stochastic delay differential equations (SDDEs) and SDDEs with pure jumps. By the Itô formula, the Taylor formula, and the Burkholder-Davis-Gundy inequality, we show that the solution of the averaged SDDEs converges to that of the standard SDDEs in the sense of pth moment and also in probability. Finally, two examples are provided to illustrate the theory.


Introduction
The averaging principle for dynamical system is important in problems of mechanics, control and many other areas. It was first put forward by Krylov and Bogolyubov [], then by Gikhman []andV olosov[] for non-linear ordinary differential equations. With the developing of the theory of stochastic analysis, many authors began to study the averaging principle for stochastic differential equations (SDEs). See, for instance, Khasminskii  On the other hand, in real phenomena, many real systems may be perturbed by abrupt pulses or extreme events and these systems with white noise are not always appropriate to interpret real data in a reasonable way. A more natural mathematical framework for these phenomena has been taken into account other than purely Brownian perturbations. It is recognized that SDEs with jumps are quite suitable to describe such discontinuous systems. In the meantime, the averaging method for a class of stochastic equations with jumps has received much attention from many authors and there exists some literature [-] concerned with the averaging method for SDEs with jumps.
Motivated by the above discussion, in this paper we study the averaging principle for a class of stochastic delay differential equations (SDDEs) with variable delays and jumps. To the best of our knowledge, the published papers on the averaging method are concentrated on the case of SDEs, there are few results concerning the averaging principle for SDDEswithjumps.Whatismore,associatedwithalltheworkmentionedabove,wepay special attention to the fact that most authors just focus on the mean-square convergence of the solution of the averaged stochastic equations and that of the standard stochastic equations. They do not consider the general pth (p > ) moment convergence case. In order to close this gap, the main aim of this paper is to study the solution of the averaged SDDEs converging to that of the standard SDDEs in the sense of pth moment. By using the Itô formula, the Taylor formula, and stochastic inequalities, we give the proof of the pth moment convergence results. It should be pointed out that we will not get the pth moment convergence results by using the proof of [-] and we need to develop several new techniques to deal with the pth moment case and the term with Poisson random measure. The results obtained are a generalization and improvement of some results in [-].
The rest of this paper is organized as follows. In Section , we prove that the solution of the averaged SDDEs converges to that of the standard SDDEs in the sense of pth moment and also in probability; in Section  we also consider the above results as regards the pure jump case. Finally, we give two examples to illustrate the theory in Section .
2 Averaging principle for Brownian motion case Let ( , F, P) be a complete probability space equipped with some filtration (F t ) t≥ satisfying the usual conditions. Here w(t)isanm-dimensional Brownian motion defined on the probability space ( , F, P) adapted to the filtration (F t ) t≥ .Letτ >andC([-τ ,];R n )denote the family of all continuous functions ϕ from [-τ ,]→ R n .ThespaceC([-τ ,];R n ) is assumed to be equipped with the norm ϕ = sup -τ ≤t≤ |ϕ(t)|.
Consider the following SDDEs: where f :[,T]×R n ×R n → R n and g :[,T]×R n ×R n → R n×m are both Borel-measurable functions. The function δ :[,T] → R is the time delay which satisfies -τ ≤ δ(t) ≤ t.The initial condition x  is defined by that is, ξ is an F  -measurable C([-τ ,];R n )-valued random variable and E ξ p < ∞.
To study the averaging method of (), we need the following assumptions. (H.) For all x  , y  , x  , y  ∈ R n and t ∈ [, T], there exist two positive constants k  and k  such that Clearly, condition ()togetherwith() implies the linear growth condition In fact, we have, for any x, y ∈ R n , Similar to the above derivation, we have which implies condition (). Similartotheproofof[], we have the following existence result.
Theorem . Under condition (H.), () has a unique solution in L p , p ≥ . Moreover, we have The proof of Theorem . is given in the Appendix. Next, let us consider the standard form of SDDEs (), where the coefficients f , g have the same conditions as in (), ()andε ∈ [, ε  ]isapositive small parameter with ε  is a fixed number.
Letf (x, y):R n × R n → R n andḡ(x, y):R n × R n → R n×m be measurable functions, satisfying condition (H.). We also assume that the following conditions are satisfied.
(H.) For any x, y ∈ R n , there exist two positive bounded functions ϕ i (T  ), i =,,such that where a = f , g and lim T  →∞ ϕ i (T  )=. Then we have the averaging form of the corresponding standard SDDEs Obviously, under condition (H.), the standard SDDEs () and the averaged SDDEs () has a unique solution on t ∈ [, T], respectively. Now, we present and prove our main results, which are used for revealing the relationship between the x ε (t)andy ε (t).
Theorem . Let the conditions (H.), (H.) hold. For a given arbitrary small number δ  >,there exist L >,ε  ∈ (, ε  ], and β ∈ (, ) such that Proof For simplicity, denote the difference e ε (t)=x ε (t)-y ε (t). From ()and(), we have Using the basic inequality ab ≤ a  + b  and taking expectation on both sides of (), it follows that, for any u ∈ [, T], By the Young inequality, it follows that for any ǫ  > where the second term of ()canbewrittenby For any ǫ  >,wederivethat Inserting ()into(), we have By setting ǫ  =+ √ k  ,weget From condition (H.), we then see that Next, we will estimate I  of (). Using the Young inequality again, it follows that for any Similartothecomputationof(), we can obtain Letting ǫ  =(+ √ k  )  ,weget From condition (H.), it follows that On the other hand, by the Burkholder-Davis-Gundy inequality, we have Similar to the estimation of I  ,wederivethat Hence, combing (), (), and (), By Theorem .,wehavethefollowingfact:foreacht ≥ , if E ξ p < ∞,thenE|y ε (t)| p < ∞. Hence condition (H.) implies that Finally, by the Gronwall inequality, we have Choose β ∈ (, ) and L >  such that for every t ∈ [, The proof is completed.
With Theorem ., it is easy to show the convergence in probability between x ε (t)and y ε (t).
Corollary . Let the conditions (H.) and (H.) hold. For a given arbitrary small number δ  >,there exists ε  ∈ [, ε  ] such that for all ε ∈ (, ε  ], we have where L and β are defined by Theorem ..
Proof By Theorem . and the Chebyshev inequality, for any given number δ  >,wecan obtain P sup Let ε → , and the required result follows.

Averaging principle for pure jump case
In this section we turn to the counterpart for SDDEs with jumps. We further need to introduce some notations. In this section, we consider the SDDEs with pure jumps: where f :[  ,T] × R n × R n → R n and h :[  ,T] × R n × R n × Z → R n are both Borelmeasurable functions. The initial condition x  is defined by To guarantee the existence and uniqueness of the solution, we introduce the following conditions on the jump term.
(H.) For all x  , y  , x  , y  ∈ R n and v ∈ Z, there exist two positive constants k  and k  such that with Z |v| p π(dv)<∞. Let us consider the standard form of SDDEs with pure jumps (), where the coefficients f , h have the same conditions as in (H.), (H.), and (H.) and ε ∈ [, ε  ] is a positive small parameter with ε  is a fixed number.
Letf (x, y):R n ×R n → R n andh(x, y, v):R n ×R n ×Z → R n be measurable functions, satisfying conditions (H.), (H.), and (H.). We also assume that the following inequalities are satisfied.
(H.) For any x, y ∈ R n and v ∈ Z, there exists a positive bounded function ϕ  (T  ), such that Then the averaging form of ()isgivenby Proof For simplicity, denote the difference e ε (t)=x ε (t)-y ε (t). From ()and(), we have By the Itô formula (see [, ]), we obtain Using the basic inequality ab ≤ a  + b  and taking expectations on both sides of (), it follows that By the Burkholder-Davis-Gundy inequality, there exists a positive constant c p such that Next, the Young inequality implies that where ǫ  > . By setting ǫ  = c - p ,weget Similar to the estimate of I  ,wehave c  =p( + k  )uϕ  (u)+(c  +c  )uϕ  (u).
By Theorem ., we have the following fact: By the Gronwall inequality, we obtain Consequently, given any number The proof is completed.
Similarly, we have the following results as regards the convergence in probability between x ε (t)andy ε (t).
Remark . When the time delay δ(t)=t,() will reduce to SDEs with jumps, which have been studied by [-]. In particularly, if p =in(), then we have the mean-square sense convergence of the standard solution of () and the averaged solution of (). So the corresponding results in [-] are generalized and improved.
Remark . In [], Tan and Lei studied the averaging method for SDDEs under non-Lipschitz conditions. In particular, we see that the Lipschitz condition is a special case of non-Lipschitz conditions which are studied by many scholars [-]. Similarly, by applying the proof of Theorem ., we can prove the standard solution of ()converges to the averaged solution of ()i nt h epth moment under non-Lipschitz conditions. In other words, we obtain a more general result on the averaging principle for SDDEs with jumps than Theorem ..

4E x a m p l e s
In this section, we construct two examples to demonstrate the averaging principle results.
Example . LetÑ(dt, dv) be a compensated Poisson random measures and is given by π(du) dt = λf (v) dv dt,whereλ >  is a constant and is the density function of a lognormal random variable. Consider the following SDEs with pure jumps: with initial data x ε () = x  ,w h e r eδ(t)=t.H e r ef (t, x ε (t)) = sin tx ε (t)a n dh(t, Hence, we have the corresponding averaged SDEs with pure jumps Now, we impose the non-Lipschitz condition on (). (H.) For all x, y ∈ R n , v ∈ Z,andp ≥ , where ρ(·) is a concave nondecreasing function from R + to R + such that ρ() = , ρ(u)> for u >and   du ρ(u) = ∞. Let us return to (). It is easy to see that h(t, ·) is a nondecreasing, positive and concave function on [, ∞]withh(t, ) = . Moreover, by a straight computation, we have Hence, the coefficients of ()and() satisfy our condition (H.). Similar to the proof of [-], we find that ()and()haveuniquesolutionsinL p , p ≥ , respectively. SimilartotheproofofTheorem., we find that the standard solution of ()converges to the averaged solution of () in the sense of the pth moment.
where φ  (t)=exp at + t  ∞  ln( + c √ εv)-c √ εv π(dv) ds When t ∈ [τ ,τ ], the explicit solution of SDDEs with jumps is given by Repeating this procedure over the interval [τ ,τ ], [τ ,τ ], etc. we can obtain the explicit solution y ε (t)o nt h ee n t i r ei n t e rv a l[  ,T]. On the other hand, it is easy to find that the conditions of Theorems . and . are satisfied, so the solution of averaged SDDEs with jumps () will converge to that of the standard SDDEs with jumps () in the sense of the pth moment and in probability.

5C o n c l u s i o n
In this paper, we study the averaging method for SDDEs and SDDEs with pure jumps. By applying the Itô formula, the Taylor formula, and the BDG inequality, we prove that the solution of the averaged SDDEs converges to that of the standard SDDEs in the sense of the pth moment and also in probability. Finally, two examples are provided to demonstrate the proposed results.
Similarly, J  and J  can be estimated as I  . Finally, all of required assertions can be obtained in the same way as the proof of Theorems . and .. The proof is therefore complete.