Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

On the equivalence between some jumping SDEs with rough coefficients and some non-local PDEs

Nicolas Fournier and Liping Xu

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Abstract

We study some jumping SDE and the corresponding Fokker–Planck (or Kolmogorov forward) equation, which is a non-local PDE. We assume only some measurability and growth conditions on the coefficients. We prove that for any weak solution $(f_{t})_{t\in[0,T]}$ of the PDE, there exists a weak solution to the SDE of which the time marginals are given by $(f_{t})_{t\in[0,T]}$. As a corollary, we deduce that for any given initial condition, existence for the PDE is equivalent to weak existence for the SDE and uniqueness in law for the SDE implies uniqueness for the PDE. This extends some ideas of Figalli (J. Funct. Anal. 254 (2008) 109–153) concerning continuous SDEs and local PDEs.

Résumé

On étudie certaines EDS à sauts et les équations de Fokker–Planck (ou Kolmogorov progressives) correspondantes, qui sont des EDP non-locales. On suppose seulement que les coefficients sont mesurables et à croissance au plus linéaire. On montre que pour toute solution faible $(f_{t})_{t\in[0,T]}$ de l’EDP, il existe une solution faible à l’EDS, dont les lois marginales sont données par $(f_{t})_{t\in[0,T]}$. On en déduit que pour toute donnée initiale, l’existence pour l’EDP est équivalente à l’existence faible pour l’EDS, et que l’unicité en loi pour l’EDS implique l’unicité pour l’EDP. Nous étendons ainsi des idées de Figalli (J. Funct. Anal. 254 (2008) 109–153) concernant des EDS continues et des EDP locales.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 55, Number 2 (2019), 1163-1178.

Dates
Received: 26 December 2016
Revised: 1 May 2018
Accepted: 15 May 2018
First available in Project Euclid: 14 May 2019

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1557820847

Digital Object Identifier
doi:10.1214/18-AIHP914

Mathematical Reviews number (MathSciNet)
MR3949969

Zentralblatt MATH identifier
07097347

Subjects
Primary: 60H10: Stochastic ordinary differential equations [See also 34F05] 60J75: Jump processes 40K05

Keywords
Existence and uniqueness Weak solution Jumping SDEs Non-local PDEs

Citation

Fournier, Nicolas; Xu, Liping. On the equivalence between some jumping SDEs with rough coefficients and some non-local PDEs. Ann. Inst. H. Poincaré Probab. Statist. 55 (2019), no. 2, 1163--1178. doi:10.1214/18-AIHP914. https://projecteuclid.org/euclid.aihp/1557820847


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