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

Cristina Anton

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arXiv · 2405.19482

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

Cristina Anton

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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.

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Cristina Anton. 2024-05-29. Malliavian differentiablity and smoothness of density for SDES with locally Lipschitz coefficients. https://arxiv.org/abs/2405.19482

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