PUB - Publikationen an der Universität Bielefeld

In this paper we show the weak differentiability of the unique strong solution with respect to the starting point x as well as Bismut-Elworthy-Li's derivative formula for the following stochastic differential equation in R-d : dX(t) = b(t, X-t)dt + sigma(t, X-t)dW(t), X-0 =x, where sigma is bounded, uniformly continuous and nondegenerate, b is an element of (L) over tilde (p1)(q1) and del sigma is an element of (L) over tilde (p2)(q2) for some Pi, qi is an element of [2, infinity) with d/pi + 2/qi < 1, i = 1, 2, where <(L)over tilde>(pi)(qi) (,) i = 1, 2 are some localized spaces of L-qi (R+; L-pi (R-d)). Moreover, in the endpoint case b is an element of (L) over tilde (d;uni)(infinity) subset of (L) over tilde (d)(infinity), we also show the weak well-posedness. (C) 2020 Elsevier B.V. All rights reserved.