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

TL;DR

This paper demonstrates that solutions to stochastic differential equations driven by fractional Brownian motion with Hurst parameter greater than 1/2 have a short-time density expansion akin to the classical Brownian motion case, under certain assumptions.

Contribution

It extends the small-time kernel expansion technique to SDEs driven by fractional Brownian motion with Hurst parameter H>1/2, a case not previously well-understood.

Findings

Density of solutions admits a short-time expansion similar to Brownian motion case.

Expansion holds under specific assumptions on the fractional Brownian motion and the SDE.

Provides a theoretical foundation for analyzing SDEs driven by fractional Brownian motion.

Abstract

In this paper we show that under some assumptions, for a $d$-dimensional fractional Brownian motion with Hurst parameter $H>1/2$, the density of solution of stochastic differential equation driven by it has a short-time expansion similar to that in the Brownian motion case.