Smooth densities of the laws of perturbed diffusion processes

Under some regularity conditions on $b$, $σ$ and $α$, 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.

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