Exponential bounds for the density of the law of the solution of a SDE with locally Lipschitz coefficients

TL;DR

This paper establishes exponential bounds and smoothness properties for the density of solutions to SDEs with locally Lipschitz coefficients under Hörmander's condition, using advanced Malliavin calculus techniques.

Contribution

It provides new exponential bounds for the density of SDE solutions with locally Lipschitz coefficients under Hörmander's hypothesis, employing Malliavin differentiability concepts.

Findings

Exponential bounds for the density established

Smoothness of the density demonstrated

Applicability to SDEs with locally Lipschitz coefficients

Abstract

Under the uniform H\"{o}rmander's hypothesis we study smoothness and exponential bounds of the density of the law of the solution of a stochastic differential equation (SDE) with locally Lipschitz drift that satisfy a monotonicity condition. To avoid non-integrability problems we use results about Malliavin differentiability based on the concepts of Ray Absolute Continuity and Stochastic Gate\^aux differentiability.