Existence, uniqueness and comparisons for BSDEs in general spaces

TL;DR

This paper develops a comprehensive theory for backward stochastic differential equations (BSDEs) in very general continuous-time spaces, establishing existence, uniqueness, and comparison results without restrictive assumptions on the filtration or martingale measures.

Contribution

It introduces a unified framework for BSDEs in general spaces, extending classical results to settings with arbitrary filtrations and enabling the construction of nonlinear expectations.

Findings

Established existence and uniqueness conditions for square-integrable solutions

Provided comparison theorems for BSDEs in general spaces

Enabled embedding of discrete processes within continuous frameworks

Abstract

We present a theory of backward stochastic differential equations in continuous time with an arbitrary filtered probability space. No assumptions are made regarding the left continuity of the filtration, of the predictable quadratic variations of martingales or of the measure integrating the driver. We present conditions for existence and uniqueness of square-integrable solutions, using Lipschitz continuity of the driver. These conditions unite the requirements for existence in continuous and discrete time and allow discrete processes to be embedded with continuous ones. We also present conditions for a comparison theorem and hence construct time consistent nonlinear expectations in these general spaces.