On the averaging principle for stochastic delay differential equations with jumps

Abstract

In this paper, we investigate the averaging principle for stochastic delay differential equations (SDDEs) and SDDEs with pure jumps. By the Itô formula, the Taylor formula, and the Burkholder-Davis-Gundy inequality, we show that the solution of the averaged SDDEs converges to that of the standard SDDEs in the sense of pth moment and also in probability. Finally, two examples are provided to illustrate the theory.

Averaging principle for a type of Caputo fractional stochastic differential equations

“…By equation (3.10), we see that, the convergence rate relate to the convergence rate function κ(•) which is different from others paper results by the similar methods as[21,22].…”

“…If we let α = 1 in equation (1.1) and (1.2), then the equation (1.1) to be a classic SDEs and the condition (1.2) is consistent with the classic averaging condition for SDEs, the averaging principle for such SDEs have been considered by many authors with similar methods, see [21,22]. It is worth pointing out that the condition lim T →∞ κ(T ) = 0 has been imposed in many papers, but it has not been used in the proofs, for details see [7,23,24].…”

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].…”

Some results in stochastic functional integro-differential equations with infinite delays

“…For further read see Bainov and Milusheva (1982); Hale (1966); Hale and Verduyn Lunel (1990); Federson and Godoy (2010) and the reference therein. Recently, few authors studied the averaging principle for stochastic differential equation under some restrictive conditions with non-Lipschitz conditions Mao et al (2015); Tan and Lei (2013). In Xu, Duan and Xu (2011), the author established averaging principle for dynamical systems with Lẽvy noise.…”