Asymptotic behaviors for distribution dependent SDEs driven by fractional Brownian motions

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 ∈ (1/2, 1) and magnitude ǫ 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.

Positivity of the Density for Rough Differential Equations

“…Study of MDPs for SDEs driven by a (mixed) fractional Brownian motion is still in its infancy. To our knowledge, there are only three works [1,7,21]. All of them are quite recent and study the case where Hurst parameter is larger than 1/2.…”