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

2015/09/24 by Priola, Enrico · 1 citation
#34F05 #60H10 #60J75 #Dynamical Systems (math.DS) #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1509.07448

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, X₀=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∈ (0,1). We assume that L₁ 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∈ (0,2) and β> 1 - α/2.

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