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

TL;DR

This paper proves the existence and uniqueness of strong solutions for time-dependent SDEs with Kato class coefficients, extending previous results to broader conditions and establishing new regularity estimates for related parabolic equations.

Contribution

It extends the theory of SDEs by establishing strong solutions under weaker Kato class conditions and includes time-dependent drifts, with new regularity estimates for parabolic equations.

Findings

Existence of unique strong solutions under Kato class conditions with  close to 2.

Extension to time-dependent drifts in SDEs.

Development of new regularity estimates for parabolic equations with 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.