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

TL;DR

This paper proves the Malliavin differentiability of solutions to certain SDEs with super-linear drift and demonstrates the smoothness of their densities under Hörmander's condition, using advanced stochastic analysis techniques.

Contribution

It establishes Malliavin differentiability of all orders for SDE solutions with polynomial growth drifts and proves density smoothness under Hörmander's hypothesis.

Findings

Solutions are Malliavin differentiable of all orders.

The density of solutions is infinitely differentiable.

Methodology avoids non-integrability issues with unbounded drifts.

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.