Small-time kernel expansion for solutions of stochastic differential equations driven by fractional Brownian motions

The goal of this paper is to show that under some assumptions, for a d -dimensional fractional Brownian motion with Hurst parameter H > 1 / 2 , the density of the solution of the stochastic differential equation X t x = x + ∑ i = 1 d ∫ 0 t V i ( X s x ) d B s i , admits the following asymptotics at small times: p ( t ; x , y ) = 1 ( t H ) d e − d 2 ( x , y ) 2 t 2 H ( ∑ i = 0 N c i ( x , y ) t 2 i H + O ( t 2 ( N + 1 ) H ) ) .