Averaging Principle for Backward Stochastic Differential Equations driven both standard and fractional Brownian motions

TL;DR

This paper develops an averaging principle for backward stochastic differential equations driven by both standard and fractional Brownian motions, enabling approximation of complex systems by simpler averaged systems.

Contribution

It introduces an averaged SFrBSDE framework and establishes conditions under which solutions can be approximated in mean square and probability.

Findings

Solutions to original SFrBSDEs can be approximated by averaged systems

The approximation holds in mean square and in probability

Provides a quantitative comparison between original and averaged solutions

Abstract

Stochastic averaging for a class of backward stochastic differential equations driven by both standard and fractional Brownian motions (SFrBSDEs in short), is investigated. An averaged SFrBSDEs for the original SFrBSDEs is proposed, and their solutions are quantitatively compared. Under some appropriate assumptions, the solutions to original systems can be approximated by the solutions to averaged stochastic systems in the sense of mean square and also in probability.