Differentiability of SDEs with drifts of super-linear growth

Abstract

We close an unexpected gap in the literature of Stochastic Differential Equations (SDEs) with drifts of super linear growth and with random coefficients, namely, we prove Malliavin and Parametric Differentiability of such SDEs. The former is shown by proving Stochastic Gâteaux Differentiability and Ray Absolute Continuity. This method enables one to take limits in probability rather than mean square or almost surely bypassing the potentially non-integrable error terms from the unbounded drift. This issue is strongly linked with the difficulties of the standard methodology of [13, Lemma 1.2.3] for this setting. Several examples illustrating the range and scope of our results are presented.

We close with parametric differentiability and recover representations linking both derivatives as well as a Bismut-Elworthy-Li formula.