An averaging principle for fractional stochastic differential equations with Lévy noise

Abstract

This paper is devoted to the study of an averaging principle for fractional stochastic differential equations in Rn with Lévy motion, using an integral transform method. We obtain a time-averaged effective equation under suitable assumptions. Furthermore, we show that the solutions of the averaged equation approach the solutions of the original equation. Our results provide a better understanding for effective approximation of fractional dynamical systems with non-Gaussian Lévy noise.

Averaging principle for a type of Caputo fractional stochastic differential equations

“…Since then, many efforts have been devoted to developing this theory for the stochastic system. Here we only highlight [7,8,9,10,11,12,13,14] and references therein.…”

“…In [19], under a different framework, when the existence and uniqueness of solutions are established, the asymptotic distance between two distinct solutions is discussed. To the best of our knowledge, the averaging principle for fractional differential equations still has a big challenge, there are only a few papers [12,20] to investigate the averaging principle for Caputo fractional SDEs. It is worth noting that the averaging principle has been obtained in these two papers by using similar methods but different assumptions.…”

An Averaging Principle for Mckean–Vlasov-Type Caputo Fractional Stochastic Differential Equations

“…Based on these ideas, a lot of effective methods have been generated in dynamical systems, such as invariant manifolds, averaging principle, and homogenization principle. ese effective methods have now been extended to deal with stochastic systems, such as stochastic invariant manifolds see [1,2] and stochastic averaging principle, see [3][4][5][6][7][8][9].…”

“…Currently, the problem of averaging for stochastic differential equations have received a lot of attention and various types of stochastic differential equations have been studied, see [4,6,7,[10][11][12]. However, there are no relevant results of averaging principle for distribution dependent-type stochastic differential equations which we will consider in this paper.…”

Averaging Principles for Nonautonomous Two-Time-Scale Stochastic Reaction-Diffusion Equations with Jump

“…Since then, the averaging principle has been an active area of research. Many studies on the averaging principle of SDEs have been presented, e.g, Givon [5], Freidlin and Wentzell [6], Duan [7], ompson [8], and Xu and his coworkers [9][10][11]. Recently, effective approximations for slow-fast SPDEs have been received extensive attention.…”