Explicit local density bounds for It\^o-processes with irregular drift

TL;DR

This paper derives explicit upper bounds for the density of continuous diffusion processes with irregular drift, extending previous results to certain one-dimensional cases with Lipschitz diffusion coefficients.

Contribution

It provides new explicit density bounds for Itô processes with irregular drift, including one-dimensional cases with Lipschitz diffusion coefficients.

Findings

Explicit upper bounds for densities of continuous diffusions with irregular drift.

Extension to one-dimensional diffusions with Lipschitz continuous diffusion coefficients.

Use of comparison to doubly reflected Brownian motion for density estimation.

Abstract

We find explicit upper bounds for the density of marginals of continuous diffusions where we assume that the diffusion coefficient is constant and the drift is solely assumed to be progressively measurable and locally bounded. In one dimension we extend our result to the case that the diffusion coefficient is a locally Lipschitz-continuous function of the state. Our approach is based on a comparison to a suitable doubly reflected Brownian motion whose density is known in a series representation.