On moments and strong local Hölder regularity of solutions of stochastic differential equations and of their spatial derivative processes

Abstract: Spatial differentiability of solutions of stochastic differential equations (SDEs) is a classical question in stochastic analysis. The case of coefficients with globally Lipschitz continuous derivatives is well understood in the literature. Counterexamples with smooth and bounded coefficients demonstrate that the non-globally Lipschitz case is more subtle. In this article we establish conditions, including a suitable local monotonicity property, which provide existence of continuously differentiable solutions of SDEs, moment estimates and strong local H\"older regularity.