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

2016/11/19 by Nicolas Fournier, Liping Xu, Fournier, Nicolas +1
Economics, Econometrics and Finance · Mathematics · Physics and Astronomy · #40K05 #60H10 #60J75 #Analysis of PDEs (math.AP) #FOS: Mathematics #Probability (math.PR) #Stochastic processes and financial applications #Stochastic processes and statistical mechanics #Theoretical and Computational Physics

paper · pdf · doi:10.48550/arxiv.1611.06341

openalex publication_date 2016/11/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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∈ [0,T]} of the PDE, there exists a weak solution to the SDE of which the time marginals are given by (f_t)_t∈[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 [5] concerning continuous SDEs and local PDEs.

Citations

• Propagation of chaos for the Landau equation with moderately soft\n potentials

Related

• Weak solutions to Kolmogorov-Fokker-Planck equations: regularity, existence and uniqueness
• Existence and Uniqueness of Stochastic PDEs associated with the Forward Equations: An Approach using Alternate Norms
• Superposition principle for non-local Fokker-Planck operators
• Fundamental solutions to Kolmogorov-Fokker-Planck equations with rough coefficients: existence, uniqueness, upper estimates
• On a Class of Stochastic Differential Equations With Jumps and Its Properties