$L^q(L^p)$-theory of stochastic differential equations

Abstract

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 ${\mathbb{R}}^d$: $$\mathrm{d}{X}_t=b(t,X_t)\mathrm{d}t+\sigma (t,X_t)\mathrm{d}{W}_t,X_0=x\in {\mathbb{R}}^d,$$ where $\sigma$ is bounded, uniformly continuous and nondegenerate, $\nabla \sigma \in {\tilde{\mathbb{L}}}_{q_1}^{p_1}$ and $b\in {\tilde{\mathbb{L}}}_{q_2}^{p_2}$ for some $p_i,q_i\in [2,\infty )$ with $\frac{d}{p_i}+\frac{2}{q_i}<1$, $i=1,2$, where ${\tilde{\mathbb{L}}}_{q_i}^{p_i},i=1,2$ are some localized spaces. Moreover, in the endpoint case $b\in {\tilde{\mathbb{L}}}_{\infty }^{d;\mathrm{uni}}$, we also show the weak well-posedness.

22pages