  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.
