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Spatial differentiability of solutions of stochastic differential equations(SDEs) is a classical question in stochastic analysis. The case of coefficientswith globally Lipschitz continuous derivatives is well understood in theliterature. Counterexamples with smooth and bounded coefficients demonstratethat the non-globally Lipschitz case is more subtle. In this article weestablish conditions, including a suitable local monotonicity property, whichprovide existence of continuously differentiable solutions of SDEs, momentestimates and strong local H\"older regularity.

On moments and strong local H\"older regularity of solutions of stochastic differential equations and of their spatial derivative processes