Smooth densities of the laws of perturbed diffusion processes

Under some regularity conditions on bb, σσ and αα, we prove that the solution of the following perturbed stochastic differential equation equation(0.1)Xt=x+∫0tb(Xs)ds+∫0tσ(Xs)dBs+αsup0≤s≤tXs,α<1 admits smooth densities for all 00t0>0 is some finite number.