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

TL;DR

This paper establishes an equivalence between certain jump stochastic differential equations with rough coefficients and their associated non-local PDEs, extending previous results to a broader class of equations.

Contribution

It proves that weak solutions of the non-local PDE correspond to weak solutions of the jump SDE, linking existence and uniqueness properties between the two.

Findings

Existence of PDE solutions implies existence of SDE solutions.

Weak uniqueness of SDE solutions implies uniqueness of PDE solutions.

Extends previous results from local to non-local PDEs with jumps.

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 [5] concerning continuous SDEs and local PDEs.