Pontryagin's optimal variational inequalities for SDEs with non-differentiable coefficients: A Malliavin calculus approach

This paper investigates necessary and sufficient optimality conditions for a specific class of controlled stochastic differential equations (SDEs) with nondegenerate diffusion matrices. We aim to extend some existing results, which relied on smooth coefficients, to the global Lipschitz setting. Key tools in our analysis include Krylov's estimate and Rademacher's theorem, which asserts that every Lipschitz function is differentiable almost everywhere with bounded Borel-measurable derivatives. The proof strategy involves approximating the original non-regular control problem through a sequence of perturbed regular control problems. We then derive the necessary and sufficient conditions for near-optimality using Malliavin calculus. Finally, by taking the limits, we obtain the desired results. Additionally, we explore an application to a class of controlled SDEs with measurable drift that exhibits quadratic growth.