Smooth density for the Solution of Scalar SDEs with Locally Lipschitz Coefficients under H\"ormander Condition

TL;DR

This paper proves the existence of a smooth probability density for solutions of certain scalar SDEs with locally Lipschitz coefficients under H"ormander's condition, using approximation and Malliavin calculus techniques.

Contribution

It establishes smooth densities for scalar SDEs with locally Lipschitz coefficients under H"ormander's condition, extending previous results to less restrictive coefficient conditions.

Findings

Proved nondegeneracy of the SDE solution under H"ormander condition.

Constructed globally Lipschitz approximations converging to the original SDE.

Used Lyapunov functions to ensure boundedness of moments and derivatives.

Abstract

In this paper the existence of a smooth density is proved for the solution of an SDE, with locally Lipschitz coefficients and semi-monotone drift, under H\"ormander condition. We prove the nondegeneracy condition for the solution of the SDE, from it an integration by parts formula would result in the Wiener space. To this end we construct a sequence of SDEs with globally Lipschitz coefficients whose solutions converge to the original one and use some Lyapunov functions to show the uniformly boundedness of the p-moments of the solutions and their Malliavin derivatives.