We obtain the unique weak and strong solvability for time inhomogeneous stochastic differential equations with the drift in subcritical Lebesgue--H\"{ o}lder spaces $L^p([0,T];{ \mathcal C}_b^{ \beta}({ \mathbb R}^d;{ \mathbb R}^d))$ and driven by $\alpha$-stable processes for $\alpha\in (0,2)$. The weak well-posedness is derived for $\beta\in (0,1)$, $\alpha+\beta>1$ and $p>\alpha/(\alpha+\beta-1)$ through Prohorov's theorem, Skorohod's representation and the regularity estimates of solutions for a class of fractional parabolic partial differential equations. The pathwise uniqueness and Davie's type uniqueness are proved for $\beta>1-\alpha/2$ by using It\^{ o}--Tanaka's trick. Moreover, we give a counterexample to the pathwise uniqueness for the supercritical Lebesgue--H\"{ o}lder drifts to explain the present result is sharp.
Categories: math.PR