Integration by Parts Formula and Smoothness of Densities of Solutions to SDEs with Locally Lipschitz Coefficients

TL;DR

This paper establishes the existence of a smooth density for solutions to certain SDEs with locally Lipschitz coefficients, using an integration by parts approach, and demonstrates exponential decay of the density and its derivatives.

Contribution

It introduces an integration by parts formula for SDEs with locally Lipschitz coefficients and proves the smoothness and decay properties of their solution densities.

Findings

Proved existence of smooth densities for solutions to SDEs with locally Lipschitz coefficients.

Derived exponential decay rates for the density and its derivatives.

Constructed an approximation scheme with globally Lipschitz drifts to facilitate analysis.

Abstract

In this work we prove the existence of a smooth density for the solution to an SDE with locally Lipschitz and semimonotone drift, and will derive an exponential decay for this density and all of its derivatives as well. Our main tool in this paper is an integration by parts formula for the solution of the mentioned SDE in the Wiener space. We construct an approximating sequence of SDEs with globally Lipschitz drifts and obtain a uniform bound for the integral of their solutions from which we derive the exponential decay for the derivatives of the density of the original SDE.