Differentiability of SDEs with drifts of super-linear growth

TL;DR

This paper proves Malliavin and parametric differentiability for SDEs with super-linear growth drifts, overcoming previous technical limitations by novel methods that allow limits in probability, with applications to derivative representations.

Contribution

It establishes differentiability results for SDEs with unbounded drifts, filling a key gap in the literature with new techniques that handle non-integrable error terms.

Findings

Proves Malliavin differentiability via Ray Absolute Continuity.

Establishes parametric differentiability and derivative representations.

Provides examples illustrating the applicability of the results.

Abstract

We close an unexpected gap in the literature of stochastic differential equations (SDEs) with drifts of super linear growth (and random coefficients), namely, we prove Malliavin and Parametric Differentiability of such SDEs. The former is shown by proving Ray Absolute Continuity and Stochastic G\^ateaux Differentiability. This method enables one to take limits in probability rather than mean square which bypasses the potentially non-integrable error terms from the unbounded drift. This issue is strongly linked with the difficulties of the standard methodology from Nualart's 2006 work, 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.