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 $dX_t = b(t, X_t)\,dt + dW_t$, $X_0=x$, 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 Lévy process $L= (L_t)$ and $b$ is $β$-Hölder continuous in the space variable, $β\in (0,1)$. We assume that $L_1$ has a finite moment of order $θ$, for some $θ>0$. Using also a new càdlàg 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 Lévy paths. We apply this result to a class of SDEs driven by non-degenerate $α$-stable Lévy processes, $α\in (0,2)$ and $β> 1 - α/2$.

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