Distribution dependent SDEs driven by fractional Brownian motions

Abstract

In this paper we study a class of distribution dependent stochastic differential equations driven by fractional Brownian motions with Hurst parameter H∈(0,1/2)∪(1/2,1). We prove the well-posedness of this type equations, and then establish a general result on the Bismut formula for the Lions derivative by using Malliavin calculus. As applications, we provide the Bismut formulas of this kind for both non-degenerate and degenerate cases, and obtain the estimates of the Lions derivative and the total variation distance between the laws of two solutions.