Smooth densities of the laws of perturbed diffusion processes

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 σ(X_s) dB_s+α\sup_{0 \le s \le t} X_s, \ \ \ α<1 \end{equation} admits smooth densities for all $0\le t\le t_0$, where $t_0>0$ is some finite number.