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

TL;DR

This paper extends Davie's type uniqueness results from Brownian-driven SDEs to those driven by Lévy processes with jumps, under certain regularity and integrability conditions, demonstrating strong solution uniqueness for a broader class of stochastic equations.

Contribution

It introduces a new approach to establish pathwise uniqueness for SDEs driven by Lévy processes with jumps, generalizing Davie's results to non-Gaussian noise with Hölder continuous coefficients.

Findings

Proves strong existence and uniqueness for SDEs with Lévy noise under Hölder continuity.

Establishes $L^p$-Lipschitz continuity of solutions with respect to initial conditions.

Demonstrates Davie's type uniqueness for almost all Lévy paths in specified conditions.

Abstract

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 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 $\beta > 1 - \alpha/2$.