Smooth densities of the laws of perturbed diffusion processes

TL;DR

This paper proves that under certain regularity conditions, solutions to a class of perturbed stochastic differential equations have smooth probability densities for all times up to a finite horizon.

Contribution

It establishes the existence of smooth densities for solutions of perturbed SDEs with a supremum term, extending previous results to this class of equations.

Findings

Solutions have smooth densities under regularity conditions.

Smooth densities exist for all times up to a finite horizon.

The results apply to SDEs with a supremum perturbation.

Abstract

Under some regularity conditions on $b$, $\sigma$ and $\alpha$, we prove that the following perturbed stochastic differential equation \begin{equation} X_t=x+\int_0^t b(X_s)ds+\int_0^t \sigma(X_s) dB_s+\alpha \sup_{0 \le s \le t} X_s, \ \ \ \alpha<1 \end{equation} admits smooth densities for all $0 \le t \le t_0$, where $t_0>0$ is some finite number.