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

2024/05/29 by Anton, Cristina
#60H07 (Primary) 60H10 #60H30 (Secondary) #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2405.19482

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 Hörmander'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 Gateâux Differentiability.

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