Davie's type uniqueness for a class of SDEs with jumps

A result of A.M. Davie [Int. Math. Res. Not. 2007] states that a multidimensional stochastic equation $d{X}_t=b(t,X_t)dt+d{W}_t$, $X_0=x$, driven by a Wiener process $W=(W_t)$ with a coefficient $b$ which is only bounded and measurable has a unique solution for almost all choices of the driving Brownian path. We consider a similar problem when $W$ is replaced by a L'evy process $L=(L_t)$ and $b$ is $\beta$-H"older continuous in the space variable, $\beta \in (0,1)$. We assume that $L_1$ has a finite moment of order $\theta$, for some $\theta >0$. Using also a new c`adl`ag regularity result for strong solutions, we prove that strong existence and uniqueness for the SDE together with $L^p$-Lipschitz continuity of the strong solution with respect to $x$ imply a Davie’s type uniqueness result for almost all choices of the L'evy paths. We apply this result to a class of SDEs driven by non-degenerate $\alpha$-stable L'evy processes, $\alpha \in (0,2)$ and $\beta > 1 - \alpha/2$.