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

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 Wiener 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 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 path. 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.

Schauder estimates for drifted fractional operators in the supercritical case

“…In the elliptic setting, when α ∈ [1, 2) and L α is a non-degenerate symmetric α-stable operator and for bounded Hölder drifts, global Schauder estimates were obtained by Priola, see e.g. Section 3 in [Pri12] and [Pri18] with respective applications to the strong well-posedness and Davie's uniqueness for the corresponding SDE. Also, when α ∈ [1, 2), elliptic Schauder estimates can be proved for more general Lévy-type generators invariant for translations, see Section 6 in [Pri18] and Remark 5.…”

Regularization effects of a noise propagating through a chain of differential equations: an almost sharp result

“…In the presentation below, we will mainly focus on Brownian driven SDEs. We can refer to the recent work of Priola[Pri17] for the more general Lévy driven case in the non-degenerate framework.…”

Gradient estimates and ergodicity for SDEs driven by multiplicative Lévy noises via coupling

“…We consider the following d-dimensional stochastic differential equation (SDE) driven by multiplicative pure jump Lévy noises Here, ν is the Lévy measure, i.e., a σ-finite measure on (R d , B(R d )) such that ν({0}) = 0 and (1 ∧ |z| 2 ) ν(dz) < ∞. Throughout this paper, we always assume that there exists a non-explosive and pathwise unique solution to SDE (1.1), see [2,3,7,17,18,19,22,29,30] for more details. We also need the following two assumptions on the coefficient σ(x):…”