Existence of spatially differentiable solutions of stochastic differential equations with non-globally monotone coefficient functions

Spatial differentiability of solutions of stochastic differential equations (SDEs) is required for the It\^o-Alekseev-Gr\"obner formula and other applications. In the literature, this differentiability is only derived if the coefficient functions of the SDE have bounded derivatives and this property is rarely satisfied in applications. In this article we establish existence of continuously differentiable solutions of SDEs whose coefficients satisfy a suitable local monotonicity property and further conditions. These conditions are satisfied by many SDEs from applications.