Strong solutions of stochastic differential equations with square integrable drift

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.