Lp(p>2)-strong convergence of an averaging principle for two-time-scales jump-diffusion stochastic differential equations

L p $L^{p }$ ( p > 2 $p>2$ )-strong convergence of multiscale integration scheme for jump-diffusion systems

“…With this time scale, the vector x t is referred to as the "slow component" and y t as the "fast component". Under suitable assumptions the authors [1,2] proved that when → 0, the slow component x t mean square converges to the solution of SDEs in the following form:…”

“…Thus a simplified equation, which is independent of the fast variable and possesses the essential features of the system, is highly desirable. On the one hand, while averaging principle [1][2][3][4][5][6] plays an important role in the research of slow component by getting a reduced equation (2), the difficulty of obtaining the effective equation (2) lies in the fact that the coefficientā(·) is given via expectation with respect to measure μ x (dy), which is usually difficult or impossible to obtain analytically, especially when the dimension m is large. On the other hand, even if we get the reduced equation, the equation cannot be solved explicitly.…”

“…Subsequently, the authors gave a multiscale integration scheme for the result in [9]. In 2015, Xu and Miao extended the result of [1] to the L p (p > 2) case under assumptions (H1)-(H5) in [2]. A natural question is as follows: Can we also establish the L p (p > 2) averaging principle by the multiscale integration scheme?…”

“…(H4) η := -(2β 3 + 2λ 2 β 4 + C g + λ 2 C h ) > 0, here β 3 and β 4 are taken from (3) and (4), λ 2 is from N t with intensity λ 2 / , C g , and C h are the Lipschitz coefficients for g and h, respectively, i.e., . It is worth pointing out that the L p (p > 2) averaging principle under assumptions (H1)-(H5) had been established in [2]. Now, we will introduce the multiscale integration scheme.…”

“…, M, where W n m are Brownian increments over a time interval δt, and N n m are Poisson increments with intensity λ 2 . Due to ergodicity [2] of the fast dynamics, we can select, among other selections, Y n 0 = y 0 . Note that the effective dynamics do not rely on .…”