THE AVERAGING METHOD FOR MULTIVALUED SDES WITH JUMPS AND NON-LIPSCHITZ COEFFICIENTS

. In this paper, we study the averaging principle for multivalued SDEs with jumps and non-Lipschitz coeﬃcients. By the Bihari’s inequality and the properties of the concave function, we prove that the solution of averaged multivalued SDE with jumps converges to that of the standard one in the sense of mean square and also in probability. Finally, two examples are presented to illustrate our theory.

As a matter of fact, Eq.(1) is called the multivalued SDEs with jumps. When h ≡ 0, it becomes an multivalued SDEs driven by continuous Brownian motion dx(t) + A(x(t))dt ∋ f (t, x(t))dt + g(t, x(t))dw(t). (2) Such multivalued SDEs was firstly introduced by Krée [1] and the existence and uniqueness of the solution has been discussed by Cépa [2]. After that, the theory of multivalued SDEs (2) has drawn increasing attention and it was studied subsequently by many authors. For example, Cépa [3], Lépingle and Marois [4], Bernardin [5], X.Zhang [6], J.Ren [7], Y.Ren [8], H.Zhang [9]. On the other hand, some scholars also studied the multivalued SDEs with jumps which is discontinuous in time and obtained a number of interesting results [10,11,12,13,14,15]. As we all know, the averaging principles is an important method which a more complicated time varying system can be approximated by an autonomous differential system. Since Krylov and Bogolyubov [16] put forward the averaging principles for dynamical systems, the averaging principles have received a lot of attention. For example, the averaging principles for determined differential equations can be found in [17,18,19,20,21]. For the averaging principles of stochastic differential equations (SDEs), we refer to [22,23,24,25,26,27,28,29,30,31]. While the theory of averaging for SDEs is well developed, the literature involving averaging principles for multivalued SDEs is scarce. Recently, Ngoran and Modeste [32] studied the averaging principle of multivalued SDEs and proved the solution of the averaged multivalued SDEs converges to that of the original multivalued SDEs. Later, Xu and Liu [33] removed the integrability condition about A in [32] and showed the convergence of the averaged multivalued SDEs and the original one. Guo and Pei [34] extended the averaging principle [33] to the case of multivalued SDEs with compensated Poisson random measures and proved that the averaged solution converges to the original solution.
However, to the best of our knowledge, there are few literature about using the averaging methods to obtain the approximate solutions to multivalued SDEs with jumps (1). In order to fill the gap, we will study the averaging principle of Eq.(1). Different from above works mentioned, the assumption given in this paper is not the classical Lipschitz condition. In fact, the global Lipschitz condition imposed on [32,33,34] is seemed to be considerably strong when one discusses variable applications in real world. For example, let us consider the multivalued SDEs with pure jumps where It is obviously that k(x) is a concave nondecreasing continuous function on R + and the coefficients f and h do not satisfy the global Lipschitz condition. In this case, the averaging principle obtained in [32,33,34] can not be applied to Eq.(3). Therefore, it is very important for us to establish the averaging principle of the multivalued SDEs with jumps (1) under some weaker conditions. In this paper, we assume that the coefficients of Eq.(1) satisfy the non-Lipschitz condition and use this condition to study the averaging principle of Eq.(1). By the Bihari's inequality and our proposed conditions, we prove that the solution of the averaged equation converges to that of the standard equation in the mean square sense. On the other hand, by lemma 3.10 and the properties of the concave function, we further give the order of the convergence for Eq.(1) in finite time interval and show the convergence in probability of the standard solution and the averaged solution. It should be pointed out that our results are also hold for multivalued SDEs without jumps (2) under non-Lipschitz conditions. The rest of this paper is organized as follows. In Section 2, we introduce some preliminaries and establish the existence and uniqueness of the solution to Eq.(1). In Section 3, we establish the averaging principle of Eq.(1). By the Bihari's inequality and some useful lemmas, we prove that the solution of the averaged equation will converge to that of the standard equation in the sense of the mean square and probability. Finally, two illustrative examples will be given in Section 4.

2.
Preliminaries and multivalued SDEs with jumps. Let (Ω, F, P ) be a complete probability space equipped with some filtration (F t ) t≥0 satisfying the usual conditions. Here w(t) is an m-dimensional F t -adapted Brownian motion. Let D([0, T ]; R n ) denote the family of all right-continuous functions with left-hand limits φ equipped with the norm ||φ|| = sup 0≤t≤T |φ(t)|. Let (R n , B(R n )) be a measurable space and π(du) a σ-finite measure on it. Let {p =p(t), t ≥ 0} be a stationary Now, we will give the definition and proposition about the multivalued maximal monotone operator. For more details, we refer to Cépa [2,3].
Denote by 2 R n for the set of all subset of R n . Let A : R n → 2 R n be a set-valued operator. Define the domain of A by and (2) A monotone operator A is called maximal monotone if and only if (4) For all α, β ∈ D([0, T ]; R n ) satisfying (α(t), β(t)) ∈ Gr(A), the measure The following lemmas are taken from [2].
In this paper, we consider the multivalued SDEs with Poisson random measure Let us consider the following assumptions.

Assumption 2.7.
A is a maximal monotone operator with D(A) = R n .

Remark 2.8. Under Assumption 2.5, we can deduce that
THE AVERAGING METHOD FOR MULTIVALUED SDEWJS 5 Remark 2.9. Let L > 0 and δ ∈ (0, 1/e) be sufficiently small. Define and where k ′ (ρ ′ ) denotes the derivative of function k(ρ). They are all concave nondecreasing functions satisfying ∫ . In particular, we see that the Lipschitz condition is a special case of our proposed condition. In other words, we obtain a more general result than that of [32,33,34].
Similar to the proof of [12,36,37], we have the following existence result.

Theorem 2.10. If Assumptions 2.5-2.7 hold. Then, there exists a unique solution
3. Stochastic averaging principle. In this section, we shall study the averaging principle for multi-valued SDEs with jumps. Let us consider the standard form of Eq.(5) with the initial value x ε (0) = x 0 . Here the coefficients f, g and h have the same conditions as in Assumptions 2.5, 2.6 and ε ∈ [0, ε 0 ] is a positive small parameter with ε 0 is a fixed number.
be measurable functions, satisfying Assumptions 2.5 and 2.6. We also assume that the following condition is satisfied.
with the initial value y ε (0) = x 0 . Obviously, under Assumptions 2.5-2.7, the standard multi-valued SDEs with jumps (6) and the averaged one (7) have a unique solution, respectively.
In order to prove our main result, we need to introduce some lemmas.
Moreover, there exists a positive constant C such that Proof. Obviously, by Theorem 38 in [38], we can easily obtained the equality (8).
By Theorem 48 in [38], we have In what follows, C > 0 is a constant which can change its value from line to line.
Proof. By the Itô formula, we have As we stated above, we can obtain that Eq. (7) has a unique solution (y ε (t),K(t)). Choose α = 0 and β = A • (0), then we have For any t 1 ∈ [0, T ], inserting (12) into (11) and taking the expectation, one gets Using the basic inequality 2ab ≤ a 2 + b 2 and Assumption 2.6, we have By the Burkholder-Davis-Gundy's inequality, the Young inequality and Assumption 2.6, we obtain

WEI MAO, LIANGJIAN HU, SURONG YOU AND XUERONG MAO
Now, we give the estimation for the last term of (13). dv). (16) Similar to (14), we have Let us estimate the second term of (16). By lemma 3.2 and the basic inequality, it follows that By the basic inequality |a + b|

THE AVERAGING METHOD FOR MULTIVALUED SDEWJS 9
Then, the Young inequality and Assumption 2.6 imply that Consequently, where C is a positive constant dependent on L 1 and L 2 . Set r(t) = 4(1 + |x 0 | 2 + ε 0 |A • (0)| 2 T )e C( √ ε+ε)t , then r(.) is the solution of the following ordinary differential equation By recurrence, it is easy to verify that Since r(t) is continuous and bounded on [0, T ], we have The proof is therefore complete. Now, we present our main results which are used for revealing the relationship between the processes x ε (t) and y ε (t).
Proof. From (6) and (7), we have By the Itô formula, we have Obviously, Lemma 2.2 implies that Taking the expectation on both sides of (22), it follows that for any u ∈ [0, T ] By the basic inequality |a + b| 2 ≤ 2|a| 2 + 2|b| 2 , we can obtain Then, by Assumptions 2.5 and 3.1, we have Similarly, we can deduce that and Now, we estimate the term Q 4 . By Assumptions 2.5, 3.1, the Burkholder-Davis-Gundy's inequality and the Young inequality, it follows that Next, by Assumptions 2.5, 3.1, lemma 3.2 and the Young inequality, we compute that It is worth noting that ε|h(s, x ε (s − ), v) −h(y ε (s), v)| 2 ≥ 0, therefore by Assumption 2.5 and Lemma 3.2, we derive that Combing with (23)-(29) together, we obtain By the Jensen inequality, this implies that } ds Letting } ds By lemma 3.4 and the boundedness of ψ i (u), i = 1, 2, 3, we have Obviously, γ(x) is a nondecreasing function on R + and γ(0) = 0. Moreover, we can obtain ∫ ) .
Noting that C(1 + C)M (ε + √ ε)u → 0 as ε → 0. Recalling the condition ∫ 0 + ds γ(s) = ∞, we can conclude that On the other hand, because G is a strictly increasing function, then we obtain that G has an inverse function which is strictly increasing, and G −1 (−∞) = 0. That is, Therefore we complete the proof.
Remark 3.7. From Theorem 3.6, we investigate the strong convergence (in momentsense) of x ε (t) to the averaging solution y ε (t) defined by (7) under non-Lipschitz condition. In other words, as long as ε is sufficiently small, then y ε (t) and x ε (t) are close enough.
Remark 3.8. If jump term h =h = 0, then equation (6) and (7) will become multivalued SDEs which have been investigated by [2,3,6,7,8,9,32,33]. Likewise, under our assumptions, we can show that the solution of the averaged multi-valued SDEs converges to that of the standard one. Moreover, even for the Wiener noise case, our result still seems to be new.
In order to prove Theorem 3.9, we need the following lemma.
2 ) is a concave nondecreasing function on R + .
Proof. Similar to lemma 2.2 in [40], we can obtain this lemma and omit its proof.
Remark 3.11. By Theorem 3.9, we can obtain that the order of convergence for Eq. (5) is about ε 1 2 −β . Moreover, we will use this theorem to show the convergence in probability between the processes x ε (t) and y ε (t). Proof. By Theorem 3.9 and the Chebyshev inequality, for any given number δ 2 > 0, we can obtain that Let ε → 0, and the required result follows.

Examples.
In this section, we will discuss two examples to illustrate our theory.

Conclusions.
In this paper, we discuss the averaging principle for the multivalued SDE with jumps under non-Lipschitz condition. By using the estimate for stochastic integral with respect to a Poisson random measure, the Bihari's inequality and the properties of the concave function, we prove that the solution of averaged multivalued SDE with jumps converges to that of the standard one in the sense of mean square and probability. Meantime, we also provide the order of convergence in finite time interval.