Malliavian differentiablity and smoothness of density for SDES with locally Lipschitz coefficients

Abstract

We study Malliavin differentiability for the solutions of a stochastic differential equation with drift of super-linear growth. Assuming we have a monotone drift with polynomial growth, we prove Malliavin differentiability of any order. As a consequence of this result, under the H\"ormander's hypothesis we prove that the density of the solution's law with respect to the Lebesgue measure is infinitely differentiable. To avoid non-integrability problems due to the unbounded drift, we follow an approach based on the concepts of Ray Absolute Continuity and Stochastic Gate\^aux Differentiability.