Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

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

Enrico Priola

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Abstract

A result of A.M. Davie (Int. Math. Res. Not. 24 (2007) rnm124) 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 $\beta$-Hölder 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 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 $\alpha$-stable Lévy processes, $\alpha\in(0,2)$ and $\beta>1-\alpha/2$.

Résumé

Un résultat de A.M. Davie (Int. Math. Res. Not. 24 (2007) rnm124) établit qu’une équation stochastique multi-dimensionnelle $dX_{t}=b(t,X_{t})\,dt+dW_{t}$, $X_{0}=x$, dirigée par un processus de Wiener $W=(W_{t})$ avec un coefficient $b$ qui est seulement borné et mesurable admet une unique solution pour presque tout choix de la trajectoire du processus $W$ la dirigeant. Nous considérons un problème similaire lorsque $W$ est remplacé par un processus de Lévy $L=(L_{t})$ et $b$ est $\beta$-Hölder continu en espace. Nous supposons que $L_{1}$ a un moment fini d’ordre $\theta$ pour un certain $\theta>0$. En utilisant un nouveau résultat de régularité càdlàg, nous prouvons que l’existence et unicité forte pour l’EDS, associées à une $L^{p}$-Lipschitz continuité de la solution forte par rapport à $x$, impliquent une unicité de type Davie pour presque tout choix de la trajectoire de Lévy. Nous appliquons ce résultat à une classe d’EDS dirigées par un processus de Lévy $\alpha$-stable non dégénéré pour $\alpha\in(0,2)$ et $\beta>1-\alpha/2$.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 54, Number 2 (2018), 694-725.

Dates
Received: 2 November 2015
Revised: 10 November 2016
Accepted: 14 December 2016
First available in Project Euclid: 25 April 2018

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1524643226

Digital Object Identifier
doi:10.1214/16-AIHP818

Mathematical Reviews number (MathSciNet)
MR3795063

Zentralblatt MATH identifier
06897965

Subjects
Primary: 60H10: Stochastic ordinary differential equations [See also 34F05] 60J75: Jump processes 34F05: Equations and systems with randomness [See also 34K50, 60H10, 93E03]

Keywords
Stochastic differential equations Lévy processes Path-by-path uniqueness Hölder continuous drift

Citation

Priola, Enrico. Davie’s type uniqueness for a class of SDEs with jumps. Ann. Inst. H. Poincaré Probab. Statist. 54 (2018), no. 2, 694--725. doi:10.1214/16-AIHP818. https://projecteuclid.org/euclid.aihp/1524643226


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