Malliavin differentiability and strong solutions for a class of SDE in Hilbert spaces

Abstract

We consider a class of Hilbert-space valued SDE’s where the drift coefficients are non- Lipschitzian in the sense of Hölder-continuity. Using a novel technique based on Malliavin calculus we show in this paper the existence and uniqueness of a mild solution to such equations. We emphasize that our approach does not rely on the Yamada-Watanabe principle. Moreover our method gives the important additional insight that the obtained solution is Malliavin differentiable - a property which was recently shown to play a crucial role in the study of the geometry of certain optimal causal transference plans, [12].