Decoupled mild solutions of path-dependent PDEs and IPDEsrepresented by BSDEs driven by cadlag martingales

TL;DR

This paper introduces a new concept of decoupled mild solutions for path-dependent PDEs and IPDEs, establishing existence, uniqueness, and representation via BSDEs driven by cadlag martingales, generalizing classical approaches.

Contribution

It generalizes the notion of mild solutions to path-dependent PDEs/IPDEs and characterizes solutions via BSDEs driven by cadlag martingales, including the identification of associated processes.

Findings

Existence and uniqueness of decoupled mild solutions established.

Representation of solutions via BSDEs driven by cadlag martingales.

Characterization of the associated processes in terms of deterministic functions.

Abstract

We focus on a class of path-dependent problems which include path-dependent (possibly Integro) PDEs, and their representation via BSDEs driven by a cadlag martingale. For those equations we introduce the notion of decoupled mild solution for which, under general assumptions, we study existence and uniqueness and its representation via the afore mentioned BSDEs. This concept generalizes a similar notion introduced by the authors in previous papers in the framework of classical PDEs and IPDEs. For every initial condition (s, $\eta$), where s is an initial time and $\eta$ an initial path, the solution of such BSDE produces a couple of processes (Y s,$\eta$ , Z s,$\eta$). In the classical (Markovian or not) literature the function u(s, $\eta$) := Y s,$\eta$ s constitutes a viscosity type solution of an associated PDE (resp. IPDE); our approach allows not only to identify u as (in our language) the unique decoupled mild solution, but also to solve quite generally the so called identification problem, i.e. to also characterize the (Z s,$\eta$) s,$\eta$ processes in term of a deterministic function v associated to the (above decoupled mild) solution u. MSC 2010 Classification. 60H30; 60H10; 35D99; 35S05; 60J35; 60J75.