Bounds for the Transition Density of Time-Homogeneous Diffusion Processes

TL;DR

This paper derives new sharp bounds for the transition densities of one-dimensional time-homogeneous diffusion processes under mild conditions, improving previous results and providing asymptotic expressions as transition time approaches zero, with extensions to multiple dimensions.

Contribution

It introduces simplified, sharp bounds for transition densities of diffusions under mild conditions, extending prior work limited to linear growth drifts and discussing multi-dimensional cases.

Findings

Derived sharp bounds for transition densities

Provided asymptotic expressions as transition time approaches zero

Extended results to multi-dimensional diffusion processes

Abstract

The paper presents new simple sharp bounds for transition density functions for time-homogeneous diffusions processes. The bounds are obtained under mild conditions on the drift and diffusion coefficients, extending and substantially improving previous results in the literature which were limited to drifts satisfying a linear growth condition. They lead to an asymptotic expression for the transition density as the transition time approaches zero. While the focus is on the one-dimensional case, an extension to multiple dimensions is discussed. Results are illustrated by numerical examples.