Stochastic averaging principle for differential equations with non-Lipschitz coefficients driven by fractional Brownian motion

Abstract

In this paper, we are concerned with the stochastic averaging principle for stochastic differential equations (SDEs) with non-Lipschitz coefficients driven by fractional Brownian motion (fBm) of the Hurst parameter [Formula: see text]. We define the stochastic integrals with respect to the fBm in the integral formulation of the SDEs as pathwise integrals and we adopt the non-Lipschitz condition proposed by Taniguchi (1992) which is a much weaker condition with wider range of applications. The averaged SDEs are established. We then use their corresponding solutions to approximate the solutions of the original SDEs both in the sense of mean square and of probability. One can find that the similar asymptotic results are suitable for those non-Lipschitz SDEs with fBm under different types of stochastic integrals.

Impulsive stochastic fractional differential equations driven by fractional Brownian motion

“…Moreover, the results are still new even when the coefficients of (1) satisfy the Lipschitz condition and under the non-Lipschitz condition used in [4], which is a particular case of our conditions. Finally, the obtained results extend and improve some published results of [1,4,28,36].…”

Averaging principle for stochastic real Ginzburg-Landau equation driven by $ \alpha $-stable process

“…Bogoliubov and Mitropolsky [2] first studied the averaging principle for the deterministic systems. The averaging principle for stochastic differential equations was proved by Khasminskii [25], see, e.g., [3,4,5,6,10,11,12,17,18,19,22,23,27,31,32,35] for further generalization. In most of the existing literature, Wiener noises are considered.…”

Averaging principle for fractional heat equations driven by stochastic measures

“…Then, Khasminskii in his seminal paper [4] studied the averaging principle for stochastic differential equations. Since then, it becomes an active research topic in the study of stochastic dynamic systems, and the averaging principle was investigated for many different types of equations, see, for example, [14,15,8,6,2,12,7,13] and the references therein. Recently, in [9], Vadym Radchenko investigated a class of stochastic heat equations driven by a stochastic measure µ, wherein the stochastic measure µ only satisfies the σ-additivity in probability (see the precise description in Section 2).…”