Strong existence and uniqueness of solutions of SDEs with time dependent Kato class coefficients

Abstract

Consider stochastic differential equations (SDEs) in $\Rd$: $dX_t=dW_t+b(t,X_t)\d t$, where $W$ is a Brownian motion, $b(\cdot ,\cdot )$ is a measurable vector field. It is known that if $|b{|}^2(\cdot ,\cdot )=|b{|}^2(\cdot )$ belongs to the Kato class $\K_{d,2}$, then there is a weak solution to the SDE. In this article we show that if $|b{|}^2$ belongs to the Kato class $\K_{d,\a}$ for some $\a \in (0,2)$ ($\a$ can be arbitrarily close to $2$), then there exists a unique strong solution to the stochastic differential equations, extending the results in the existing literature as demonstrated by examples. Furthermore, we allow the drift to be time-dependent. The new regularity estimates we established for the solutions of parabolic equations with Kato class coefficients play a crucial role.