Nonlinear SDEs driven by L\'evy processes and related PDEs

TL;DR

This paper investigates nonlinear stochastic differential equations driven by Le9vy processes, establishing existence, uniqueness, and regularity of solutions, and linking them to nonlinear PDEs involving fractional Laplacians.

Contribution

It extends classical SDE theory to Le9vy-driven equations with Lipschitz coefficients, providing new existence, uniqueness, and regularity results, and connecting solutions to fractional PDEs.

Findings

Proved existence and uniqueness of solutions for nonlinear Le9vy-driven SDEs.

Established absolute continuity of solution marginals under certain conditions.

Linked solutions to nonlinear PDEs with fractional Laplacian for symmetric stable processes.

Abstract

In this paper we study general nonlinear stochastic differential equations, where the usual Brownian motion is replaced by a L\'evy process. We also suppose that the coefficient multiplying the increments of this process is merely Lipschitz continuous and not necessarily linear in the time-marginals of the solution as is the case in the classical McKean-Vlasov model. We first study existence, uniqueness and particle approximations for these stochastic differential equations. When the driving process is a pure jump L\'evy process with a smooth but unbounded L\'evy measure, we develop a stochastic calculus of variations to prove that the time-marginals of the solutions are absolutely continuous with respect to the Lebesgue measure. In the case of a symmetric stable driving process, we deduce the existence of a function solution to a nonlinear integro-differential equation involving the fractional Laplacian.