Large deviation principle for the mean reflected stochastic differential equation with jumps

In this paper, we establish a large deviation principle for a mean reflected stochastic differential equation driven by both Brownian motion and Poisson random measure. The weak convergence method plays an important role.


Introduction
Consider the mean reflected stochastic differential equation (MR-SDE for short) described by the following system: where E = R \ {0}, b, σ , and F are Lipschitz functions from R to R, h is bi-Lipschitz continuous, N is a compensated Poisson measure N(ds, dz) = N(ds, dz)ϑ(dz) ds, and {B t } t≥0 is a standard Brownian motion independent of N . The integral of the function h with respect to the law of the solution to the SDE is asked to be nonnegative. The solution to (1) is the couple of continuous processes (X, K), where K is needed to ensure that the constraint is satisfied in a minimal way according to the last condition, namely the Skorokhod condition.
MR-SDE is a very special type of reflected stochastic differentials equations (SDEs) in which the constraint is not directly on the paths of the solution to the SDE as in the usual case but on the law of the solution. This kind of processes has been introduced recently by Briand, Elie, and Hu [4] in backward forms under the context of risk measures. Briand et al. [3] studied the MR-SDE in forward forms, and they provided an approximation of solution to the MR-SDE with the help of interacting particles systems.
Since the original work of Freidlin and Wentzell [11], the small noise large deviation principles for stochastic (partial) differential equations have been extensively studied in the literature. In this setting, one considers a small parameter multiplying the noise term and is interested in asymptotic probabilities of behavior as the parameter approaches zero. Earlier works on this family of problems relied on approximations and exponential probability estimates, see [2,9]. Later, Dupuis and Ellis [10] developed a weak convergence approach to the theory of large deviation. This approach is mainly based on some variational representation formula about the Laplace transform of bounded continuous functionals. The weak convergence approach has now been adopted for the study of large deviation problems for stochastic partial differential equations, see [8,13,14,16,17], etc. It is also used to study the moderate deviation problems for stochastic partial differential equations, see [7,12,15].
We will use the weak convergence approach to study the large deviation principle for MR-SDE. Here, a representation formula of K plays an important role to overcome the difficulty coming from the fact that the reflection process K depends on the law of the position.
The rest of this paper is organized as follows. In Sect. 2, we first give the definition of the solution to Eq. (1), and then we state the main results of this paper. The weak convergence criterion for the large deviation principle is recalled in Sect. 3. In Sect. 4, we shall prove the main result.

Framework and main results
We consider the following conditions.

Condition 2.1
(i) Lipschitz assumption: For any p > 0, there exists a constant C p > 0 such that, for all (ii) The random variable X 0 is square integrable independent of B t and N t .
where (U t ) 0≤t≤T is the process defined by Moreover, for any p ≥ 2, there exists a positive constant K p , depending on T, b, σ , F, h, such that In this paper, we are concerned with the large deviation principle (LDP for short) for MR-SDEs of jump type on R:

Condition 2.4
The function F satisfies the following: (2) There exists a function L F ∈ L 1 (ϑ) ∩ L 2 (ϑ) such that, for any ( For any δ > 0, define a class of functions

Condition 2.5
The functions M F and L F are in the class H δ for some δ > 0.
Remark 2.6 Condition 2.5 implies that, for all δ ∈ (0, ∞) and The main result of this paper is the following theorem.

Poisson random measure and Brownian motion
Recall that a Poisson random measure n on E T with intensity measure ϑ T is an N ϕ is the controlled random measure with ϕ selecting the intensity for the points at location x and time s.
For any ϕ ∈ A, the quantity is well defined as a [0, ∞]-valued random variable. Let Set U := L 2 × A. Definẽ

The weak convergence criterion
In this subsection, we recall a general criterion for a large deviation principle established in [8]. Let {G ε } ε>0 be a family of measurable maps from V to U, where V is introduced in Sect. 3.1 and U is a Polish space. We present a sufficient condition of large deviation principle for the family Z ε := G ε ( √ εB, εN ε -1 ), as ε → 0. Define This identification induces a topology on S Υ under which S Υ is a compact space, see the Appendix of [6]. Throughout we use this topology on S Υ . We also use the weak topology onS Υ . Set S Υ :=S Υ × S Υ , S := Υ ≥1 S Υ , and The following condition is sufficient for establishing an LDP for a family {Z ε } ε>0 .

Condition 3.1
There exists a measurable map G 0 : V → U such that the following conditions hold: (a) For any Υ ∈ N, let (f n , g n ), (f , g) ∈ S Υ be such that (f n , g n ) → (f , g) as n → ∞. Then where L T (q) is given by (10). By convention, Recall the following criterion from [8].
For applications, the following strengthened form of Theorem 3.2 is useful. Let {K n } n≥1 be an increasing sequence of compact sets in X such that ∞ n=1 K n = E. For each n, let A b,n := ϕ ∈ A : 1/n ≤ ϕ(·, x, ·) ≤ n if x ∈ K n ; ϕ(·, x, ·) = 1 if x ∈ K c n , To use the representation formula (2) of the process K , we recall a result from [3]. Define the function With these notations, denoting by (μ t ) t∈[0,T] the family of marginal laws of (U t ) t∈[0,T] , we have For any two measures ν and ν , the Wasserstein-1 distance between ν and ν is defined by From Remark 1 in [5], we have By the definition of G 0 (X s ) = 0, if s < t, using Lemma 4.1, we have The following lemma can be proved by using the argument in [6,Lemma 3.4], [7,Lemma 4.3]. We omit its proof. and (ii) For ever η > 0, there exists δ > 0 such that, for any A ⊂ [0, T] satisfying λ T (A) < δ, The following lemma is from [6, Lemma 3.11].

Lemma 4.3 Let k : [0, T] × E → R be a measurable function such that
and for all δ ∈ (0, ∞) and For any Υ ∈ N, let g n , g ∈ S Υ be such that g n → g as n → ∞. Then Under Conditions (A.1) and (A.2), Eq. (5) has a unique strong solution X ε . Therefore, there exists a Borel-measurable function G ε :V → D([0, T]; R) such that, for any Poisson is the unique solution of Eq. (5).
Next we introduce the map G 0 which will be used to define the rate function and also used for verification of Condition 3.1. Recall S defined in the last section. Under Condition 2.4, for every q = (f , g) ∈ S, the deterministic integral equation has a unique continuous solution. Here where (U t ) 0≤t≤T is the process defined by Let I : D([0, T]; R) → [0, ∞] be defined as in (11) with G 0 given by (22). We first verify Condition (3.1.a).
Set κ n (t) := sup u∈[0,t] |X n (u) -X(u)|. By the Lipschitz condition of b, we have By the Lipschitz condition of σ , we have By the linear growth of σ , we know that Since Thus, by the Cauchy-Schwarz inequality, we have Similarly, we have, for any 0 ≤ t 1 < t 2 ≤ T, which means that the sequence {I n 3 : n ≥ 1} is equicontinuous. By the Arzéla-Ascoli theorem, we know that {I n 3 : n ≥ 1} is relatively compact in C([0, T]; R d ). By using (26) and the fact that f n → f weakly in L 2 ([0, T]; R), we obtain that, for any t ∈ [0, T], By Condition 2.4, Remark 2.6, and Lemma 4.3, we know that as n → ∞, Recalling the definition of K q t given by (21), we have According to Lemma 4.1, we know that The proof is complete.
We now verify the second part of Condition 3.1.
T has the same law as that of ( under P V , there exists a unique solution to the following controlled stochastic evolution Here K ε t is given by with the process ( U ε t ) 0≤t≤T defined by Then The following estimate can be proved in a similar way to (4), which is omitted here.