Strong solutions of stochastic differential equations with square integrable drift

TL;DR

This paper establishes the existence, uniqueness, and regularity properties of strong solutions to stochastic differential equations with square integrable and Hölder continuous drift coefficients, including differentiability and Hölder continuity of solutions.

Contribution

It proves the existence and uniqueness of strong solutions under minimal regularity assumptions on the drift, and demonstrates their regularity and differentiability properties.

Findings

Strong solutions exist and are unique under square integrable and Hölder continuous drift.

Solutions have a continuous modification that is Hölder continuous in space.

The solutions are differentiable as $L^2$-valued functions.

Abstract

We prove the existence and uniqueness of strong solutions for stochastic differential equations in which the drift coefficient is square integrable in time variable and H\"{o}lder continuous in space variable. Moreover, we prove that the unique strong solution has a continuous modification, which is $\beta$-H\"{o}lder continuous in space variable for every $\beta\in (0,1)$, and as an $L^2(\Omega\times (0,T))$ valued function, it is differentiable as well.