Asymptotic behaviors for distribution dependent SDEs driven by fractional Brownian motions

TL;DR

This paper investigates the small-time asymptotic behaviors of distribution dependent SDEs driven by fractional Brownian motions, establishing large and moderate deviation principles and a central limit theorem.

Contribution

It develops a variational framework and weak convergence criteria specific to fractional Brownian motion to derive deviation principles and a CLT for these equations.

Findings

Established large deviation principles for the equations.

Proved moderate deviation principles.

Derived a central limit theorem involving Lions derivative.

Abstract

In this paper, we study small-time asymptotic behaviors for a class of distribution dependent stochastic differential equations driven by fractional Brownian motions with Hurst parameter $H\in(1/2,1)$ and magnitude $\ep^H$. By building up a variational framework and two weak convergence criteria in the factional Brownian motion setting, we establish the large and moderate deviation principles for this type equations. Besides, we also obtain the central limit theorem, in which the limit process solves a linear equation involving the Lions derivative of the drift coefficient.