Short time kernel asymptotics for Young SDE by means of Watanabe distribution theory

TL;DR

This paper investigates the short time behavior of the density function for solutions to SDEs driven by fractional Brownian motion with Hurst parameter greater than 1/2, using advanced Malliavin calculus techniques.

Contribution

It provides new short time asymptotic results for densities of fractional SDEs employing Watanabe distribution theory under ellipticity conditions.

Findings

Established on-diagonal asymptotics for the density.

Derived off-diagonal asymptotics under mild assumptions.

Applied Watanabe's theory to fractional Brownian motion driven SDEs.

Abstract

In this paper we study short time asymptotics of a density function of the solution of a stochastic differential equation driven by fractional Brownian motion with Hurst parameter $H \in (1/2, 1)$ when the coefficient vector fields satisfy an ellipticity condition at the starting point. We prove both on-diagonal and off-diagonal asymptotics under mild additional assumptions. Our main tool is Malliavin calculus, in particular, Watanabe's theory of generalized Wiener functionals.