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 required for the Itô-Alekseev-Gröbner 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.

A stochastic Gronwall inequality and applications to moments, strong completeness, strong local Lipschitz continuity, and perturbations

“…In particular, we considerably improve existing results in the literature on these applications; see Section 3 below for details. Moreover, in the subsequent article Hudde et al [22] we apply Theorem 2.4 and Corollary 2.5 to derive versions of solutions of SDEs which are twice continuously differentiable in the initial value without assuming the coefficients of the SDE to satisfy a global monotonicity condition.…”

On the strong regularity of degenerate additive noise driven stochastic differential equations with respect to their initial values

“…The regularity analysis of nonlinear stochastic differential equations (SDEs) with respect to their initial values is an active research topic in stochastic analysis (cf., e.g., [2,3,5,7,9,10,15,18,19,20,25] and the references mentioned therein). In particular, it has recently been revealed in the literature that there exist SDEs with smooth coefficient functions which have very poor regularity properties in the initial value.…”