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.