Forward-backward SDEs with distributional coefficients

TL;DR

This paper introduces two new classes of multidimensional forward-backward stochastic differential equations with distributional coefficients, establishing existence, uniqueness, and links to PDEs via a nonlinear Feynman-Kac formula.

Contribution

It develops a new framework for FBSDEs with distributional coefficients, proving existence and uniqueness results and connecting them to semi-linear PDEs involving such coefficients.

Findings

Existence and uniqueness of solutions for the first FBSDE.

Weak existence for the second FBSDE.

Establishment of a nonlinear Feynman-Kac representation for the associated PDE.

Abstract

Forward-backward stochastic differential equations (FBSDEs) have attracted significant attention since they were introduced almost 30 years ago, due to their wide range of applications, from solving non-linear PDEs to pricing American-type options. Here, we consider two new classes of multidimensional FBSDEs with distributional coefficients (elements of a Sobolev space with negative order). We introduce a suitable notion of a solution, show existence and uniqueness of a strong solution of the first FBSDE, and weak existence for the second. We establish a link with PDE theory via a nonlinear Feynman-Kac representation formula. The associated semi-linear second order parabolic PDE is the same for both FBSDEs, also involves distributional coefficients and has not previously been investigated; our analysis uses mild solutions, Sobolev spaces and semigroup theory.