On the averaging principle for SDEs driven by G-Brownian motion with non-Lipschitz coefficients

Abstract

In this paper, we aim to develop the averaging principle for stochastic differential equations driven by G-Brownian motion (G-SDEs for short) with non-Lipschitz coefficients. By the properties of G-Brownian motion and stochastic inequality, we prove that the solution of the averaged G-SDEs converges to that of the standard one in the mean-square sense and also in capacity. Finally, two examples are presented to illustrate our theory.

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.…”

Time-averaging principle for G-SDEs based on Lyapunov condition

“…As the reviewer pointed out to the author, the global Lipschitz is not essential, the main difference is that the author obtained a strong convergence result instead of weak convergence compared with Hu et al [9], but the author needed quite a strong condition (B). Mao, Chen, and You [15] obtained an excellent result about the averaging principle for stochastic differential equations driven by G-Brownian motion with global non-Lipschitz coefficients. Recent important progress in the theory of volatility uncertainty/G-Brownian motion is reviewed by Peng [16] with comments on its explanation, theory, and significance.…”

Fractional Stochastic Differential Equations Driven By G-Brownian Motion with Delays

“…The averaging method is a powerful tool to strike a balance between complex models that are more realistic and simpler models that are more amenable to analysis and simulation. For the averaging principle for FSDEs we refer to [1,7,13,14].…”