Weak order in averaging principle for stochastic differential equations with jumps

The present article deals with the averaging principle for a two-time-scale system of jump-diffusion stochastic differential equation. Under suitable conditions, the weak error is expanded in powers of timescale parameter. It is proved that the rate of weak convergence to the averaged dynamics is of order $1$. This reveals the rate of weak convergence is essentially twice that of strong convergence.


Introduction
We consider a two-time-scale system of jump-diffusion stochastic differential equation in form of dX ǫ t = a(X ǫ t , Y ǫ t )dt + b(X ǫ t )dB t + c(X ǫ t− )dP t , X ǫ 0 = x, (1.1) where X ǫ t ∈ R n , Y ǫ t ∈ R m , the drift functions a(x, y) ∈ R n , f (x, y) ∈ R m , the diffusion functions b(x) ∈ R n×d1 , c(x) ∈ R n , g(x, y) ∈ R m×d2 and h(x, y) ∈ R m . B t and W t are the vectors of d 1 , d 2 -dimensional independent Brownian motions on a complete stochastic base (Ω, F , F t , P), respectively. P t is a scalar simple Poisson process with intensity λ 1 , and N ǫ t is a scalar simple Poisson process with intensity λ2 ǫ . The positive parameter ǫ is small and describes the ratio of time scales between X ǫ t and Y ǫ t . Systems (1.1)-(1.2) with two time scales occur frequently in applications including chemical kinetics, signal processing, complex fluids and financial engineering.
With the separation of time scale, we can view the state variable of the system as being divided into two parts: the "slow" variable X ǫ t and the "fast" variable Y ǫ t . It is often the case that we are interested only in the dynamics of slow component. Then a simplified equation, which is independent of fast variable and possesses the essential features of the system, is highly desirable. Such a simplified equation is often constructed by averaging procedure as in [2,20] for deterministic ordinary differential equations, as well as the further development [13,7,8,18,19,14,15,16,25] for stochastic differential equations with continuous Gaussian processes. As far as averaging for stochastic dynamical systems in infinite dimensional space is concerned, it is worthwhile to quote the important works of [4,5,6,26] and also the works of [9,10,21]. For related works on averaging for multivalued stochastic differential equations we refer the reader to [12,22].
In order to derive the averaged dynamics of the system (1.1)-(1.2), we introduce the fast motion equation with a frozen slow component x ∈ R n in form of 3) whose solution is denoted by Y ǫ t (y). Under suitable conditions on f, g and h, Y ǫ t (y) induces a unique invariant measure µ x (dy) on R m , which is ergodic and ensures the averaged equation: where the averaging nonlinearity is defined by settinḡ In [11], it was shown that under the above conditions the slow motion X ǫ t converges strongly to the solutionX t of the above averaged equation with jumps. The order of convergence 1 2 in strong sense was provided in [17]. To our best knowledge, there is no existing literature to address the weak order in averaging principle for jump diffusion stochastic differential systems. In fact, it is fair to say that the weak convergence in stochastic averaging theory of systems driven by jump noise is not fully developed yet, although some strong approximation results on the rate of strong convergence were obtained [1,23,24].
Therefore, we aim to study this problem in this paper. Here we are interested in the rate of weak convergence of the averaging dynamics to the true solution of slow motion X ǫ t . In other word, we will determine the order, with respect to timescale parameter ǫ, of weak deviation between original solution to slow equation and the solution of the corresponding averaged equation. The main technique we adapted is to find an expansion with respect to ǫ of the solutions of the Kolmogorov equations associated with the jump diffusion system. The solvability of the Poisson equation associated with the generator of frozen equation provides an expression for the coefficients of the expansion. As a result, the boundedness for the coefficients of expansion can be proved by smoothing effect of the corresponding transition semigroup in the space of bounded and uniformly continuous functions, where some regular conditions is needed on drift and diffusion term.
Our result shows that the weak convergence rate to be 1 even when there are jump components in the system. It is the main contribution of this work. We would like to stress that asymptotic method was first applied by Bréhier [3] to an averaging result for stochastic reaction-diffusion equations in the case of Gaussian noise of additive type was included only in the fast motion. However, the extension of this argument is not straightforward. The method used in the proof of weak order in [3] is strictly related to the differentiability in time of averaged process. Therefore, once the noise is introduced in the slow equation, difficulties will arise and the procedure becomes more complicated. Our result in this paper bridges such a gap, in which the slow and the fast motions are both perturbed by noise with jumps.
The rest of the paper is structured as follows. Section 2 is devoted to notations, assumptions and summarize preliminary results. The ergodicity of fast process and the averaged dynamics of system with jumps is introduced in Section 3. Then the main result of this article, which is derived via the asymptotic expansions and uniform error estimates, is presented in Section 4. Finally, we give the appendix in section 5.
It should be pointed out that the letter C below with or without subscripts will denote generic positive constants independent of ǫ in the whole paper.

Assumptions and preliminary results
For any integer d, the scalar product and norm on d−dimensional Euclidean space R d are denoted by ·, · R d and · R d , respectively. For any integer k, we denote by C k b (R d , R) the space of all k−times differentiable functions on R d , which have bounded uniformly continuous derivatives up to the k-th order.
In what follows, we shall assume that the drift and diffusion coefficients arising in the system fulfill the following conditions. (A1) The mappings a(x, y), b(x), c(x), f (x, y), g(x, y) and h(x, y) are of class C 2 and have bounded first and second derivatives. Moreover, we assume that a(x, y), b(x) and c(x) are bounded.
(A2) There exists a constant α > 0 such that for any x ∈ R n , y ∈ R m it holds y T g(x, y)g T (x, y)y ≥ α y R m .
(A3) There exists a constant β > 0 such that for any y 1 , y 2 ∈ R m and x ∈ R n it holds Remark 2.1 Notice that from (A1) it immediately follows that the following directional derivatives exist and are controlled: where L is a constant independent of x, y, k 1 , k 2 , l 1 and l 2 . For differentiability of mappings b, c, f, g and h we possess the analogous results. For examples, we have As far as the assumption (A2) is concerned, it is a sort of non-degeneracy condition and it is assumed in order to have the regularizing effect of the Markov transition semigroup associated with the fast dynamics. Assumption (A3) is the dissipative condition which determines how the fast equation converges to its equilibrium state.
As assumption (A1) holds, for any ǫ > 0 and any initial conditions x ∈ R n and y ∈ R m , system (1.1)-(1.2) admits a unique solution, which, in order to emphasize the dependence on the initial data, is denoted by (X ǫ t (x, y), Y ǫ t (x, y)). Moreover the following lemma holds (for a proof see e.g. [17]).

Frozen equation and averaged equation
Fixing ǫ = 1, we consider the fast equation with frozen slow component x ∈ R n , Under assumptions (A1)-(A3), such a problem has a unique solution, which satisfies [17]: Moreover, as discussion in [17] and [11], equation (3.1) admits a unique ergodic invariant measure µ x satisfying Then, by averaging the coefficient a with respect to the invariant measure µ x , we can define an R n -valued mappinḡ Due to assumption (A1), it is easily to check thatā(x) is 2-times differentiable with bounded derivatives, and hence it is Lipschitz-continuous such that According to invariant property of µ x , (3.4) and assumption (A1), we have Now we can introduce the effective dynamical system As the coefficientsā, b and c are Lipschitz-continuous, this equation admits a unique solution such that With the above assumptions and notations we have the following result, which is a direct consequence of Lemma 4.1, Lemma 4.2 and Lemma 4.5.
Theorem 3.1 Assume that x ∈ R n and y ∈ R m , Then, under assumptions (A1), (A2) and (A3), for any T > 0 and φ ∈ C 3 b (R n , R), there exists a constant C T,φ,x,y such that As a consequence, it can be claimed that the weak order in averaging principle for jump-diffusion stochastic systems is 1.

Asymptotic expansion
We are now ready to seek an expansion formula for u ǫ (t, x, y) with respect to ǫ with the form where u 0 and u 1 are smooth functions which will be constructed below, and r ǫ is the remainder term. To this end, let us recall the Kolmogorov operator corresponding to the slow motion equation, with a frozen fast component y ∈ R m , which is a second order operator taking form For any frozen slow component x ∈ R m , the Kolmogorov operator for equation (3.1) is given by We set It is known u ǫ (t, x, y) solves the equation Also recall the Kolmogorov operator associated with the averaged equation (3.6) is defined as

The leading term
Let us begin with constructing the leading term. By substituting expansion (4.1) into (4.2), we see that By equating powers of ǫ, we obtain the following system of equations: According to (4.4), we can conclude u 0 does not depend on y, that is u 0 (t, x, y) = u 0 (t, x).
We also impose the initial condition u 0 (0, x) = φ(x). Note that L 2 is the generator of a Markov process defined by equation (3.1), which admits a unique invariant measure µ x , we have Thanks to (4.5), this yields so that u 0 andū are described by the same evolutionary equation. By uniqueness argument, we easily have the following lemma:

Construction of u 1
According to Lemma 4.1, (4.3) and (4.5), we get which means that where ρ is of class C 2 with respect to y, with uniformly bounded derivatives. Moreover, for any t ≥ 0 and x ∈ R n , the equality (4.6) guarantees that R m ρ(t, x, y)µ x (dy) = 0.
For any y ∈ R m and s > 0 we have here P s [ρ(t, x, y)] := Eρ(t, x, Y x s (y)). Recalling that µ x is the unique invariant measure corresponding to Markov process Y x t (y) defined by equation (3.1), from Lemma 5.1 we infer that

Now it follows from (3.3) and (3.4) that
With the aid of the above limit, we can deduce from (4.8) that  is the solution to equation (4.7).

Determination of remainder r ǫ
We now turn to the construction for remainder term r ǫ . It is known that which, together with (4.4) and (4.5), implies In order to estimate the remainder term r ǫ we need the following two lemmas.

Lemma 4.3
Under assumptions (A1), (A2) and (A3), for any x ∈ R n , y ∈ R m and T > 0, we have Proof In view of (4.9), we get By Lemma 5.6 introduced in Section 5, we have so that from (3.5) we have

Lemma 4.4
Under assumptions (A1), (A2) and (A3), for any x ∈ R n , y ∈ R m and T > 0, we have Proof Recalling that u 1 (t, x, y) is the solution of equation (4.7) and equality (4.9) holds, we have (4.12) and then, in order to prove the boundedness of L 1 u 1 , we have to estimate the three terms arising in the right hand side of above equality.
Step 1: Estimate of a(x, y), D x u 1 (t, x, y) R n . For any k ∈ R n , we have x, y, k) + I 2 (t, x, y, k).

By Lemma 5.1 and 5.4, we infer that
By Lemma 5.2 and inequality (3.5), we obtain This, together with (4.13), implies and then, as a(x, y) is bounded, it follows Step 2: Estimate of T r D 2 xx u 1 (t, x, y) · b(x)b T (x) . Since u 1 (t, x, y) is given by the representation formula (4.9), for any k 1 , k 2 ∈ R n we have Thanks to Lemma 5.1 and Lemma 5.5 we get (4.14) By Lemma 5.4 and (3.5) we infer that With a similar argument we can also show that By making use of Lemma 5.3 and (3.5), we get In view of the above estimates (4.14), (4.15), (4.16) and (4.17), we can conclude that there exists a constant C T such that which means that for fixed y ∈ R m and t ∈ [0, T ], where · L(R n ,R) denotes the usual operator norm on Banach space consisting of bounded and linear operators from R n to R. As the diffusion function g is bounded, we get Step 3: Estimate of λ 1 [u 1 (t, x + c(x), y) − u 1 (t, x, y)]. By Lemma 4.2 and boundedness condition of c(x), we directly have Finally, it is now easy to gather all previous estimates for terms in (4.12) and conclude Proof By a variation of constant formula, we write the equation (4.11) in its integral form Since u ǫ andū satisfy the same initial condition, we have so that, thanks to (4.10), (2.1) and (2.2) we have

Using Lemma 4.3 and Lemma 4.4 yields
and, according to (2.1) and (2.2), this implies that The last inequality together with (4.18) yields

Appendix
In this appendix we collect some technical results to which we appeal in the proofs of the main results in Section 4 .
Lemma 5.1 For any T > 0, there exists a constant C T > 0 such that for any x, k ∈ R n and t ∈ [0, T ], we have Proof Observe that for any k ∈ R n , where η k,x t denotes the first mean-square derivative ofX t (x) with respect to x ∈ R n along the direction k ∈ R n , then we have This means that η k,x t is the solution of the integral equation and then thanks to assumption (A1), we get Then by Gronwall lemma it follows that Next, we introduce an analogous result for the second derivative ofū(t, x).
Lemma 5.2 For any T > 0, there exists a constant C T > 0 such that for any x, k 1 , k 2 ∈ R n and t ∈ [0, T ], we have where ξ k1,k2,x t is the solution of the second variation equation corresponding to the averaged equation, which may be rewritten in the following form: Thus, by assumption (A1) and (5.1) we have By applying the Gronwall lemma we have By using the analogous arguments used before, we can prove the following estimate for the third order derivative ofū(t, x) with respect to x.

Lemma 5.3
For any T > 0, there exists a constant C T > 0 such that for any x, k 1 , k 2 , k 3 ∈ R n and t ∈ [0, T ], we have The following lemma states boundedness for the first derivative ofā(x) − Ea(x, Y x t (y)) with respect to x.
Lemma 5.4 There exists a constant C > 0 such that for any x ∈ R n , y ∈ R m , k ∈ R n and t > 0 it holds Proof The proof is a modification of the proof of [3, Proposition C.2]. For any t 0 > 0, we setã Then we have By Markov property, we havẽ Due to assumption (A1), for any k ∈ R n we have where the symbolsâ ′ x andâ ′ y denote the directional derivatives with respect to x and y, respectively. Note that the first derivative ζ x,y,k t = D x Y x t (y) · k, at the point x and along the direction k ∈ R n , is the solution of equation with initial data ζ x,y,k 0 = 0. Hence, by assumption (A1), it is straightforward to check E ζ x,y,k t R m ≤ C k R n (5.4) for any t ≥ 0. Note that for any y 1 , y 2 ∈ R m , we have where (3.3) was used to obtain the last inequality. This means that â ′ y (x, y, t) · l R m ≤ Ce − β 2 t l R m , l ∈ R m . (5.5) From (5.4) and (5.5), we obtain For the third term, by making use of assumption (A1) again, we can infer that Now, returning to (5.7) and taking into account of (5.8), (5.9) and (5.10), we get which leads to where we used the inequality (3.2). Returning to (5.3), by (5.6) and (5.11) we conclude that D xãt0 (x, y, t) · k R n ≤ Ce − β 2 t (1 + x R n + y R m ) k R n .
Taking the limit as t 0 → +∞ we obtain D x (ā(x) − Ea(x, Y x t (y))) R n ≤ Ce − β 2 t k R n (1 + x R n + y R m ) .
Proceeding with similar arguments above we obtain the following higher order differentiability.
Lemma 5.5 There exists a constant C > 0 such that for any x, k 1 , k 2 ∈ R n , y ∈ R m and t > 0 it holds D 2 xx (ā(x) − Ea(x, Y x t (y)))(k 1 , k 2 ) R n ≤ Ce − β 2 t k 1 R n k 2 R n (1 + x R n + y R m ) .
Finally, we introduce the following auxiliary result.
Lemma 5.6 There exists a constant C > 0 such that for any x, k ∈ R n , y ∈ R m and t > 0 it holds ∂ ∂t D xū (t, x) · k R n ≤ C k R n .