# Smooth densities of the laws of perturbed diffusion processes

Research paper by Lihu Xu, Wen Yue, Tusheng Zhang

Indexed on: 23 Jan '16Published on: 23 Jan '16Published in: Mathematics - Probability

#### Abstract

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