Quadratic BSDEs with jumps: a fixed-point approach

TL;DR

This paper establishes the existence and uniqueness of bounded solutions for quadratic backward stochastic differential equations with jumps using a fixed-point method, extending previous results to more general settings.

Contribution

It introduces a fixed-point approach to prove existence and uniqueness of solutions for quadratic BSDEs with jumps, including stability and comparison results.

Findings

Existence of bounded solutions under small terminal conditions

Extension to general bounded solutions via splitting technique

Stability and comparison theorems under additional assumptions

Abstract

In this article, we prove the existence of bounded solutions of quadratic backward SDEs with jumps, that is to say for which the generator has quadratic growth in the variables (z,u). From a technical point of view, we use a direct fixed point approach as in Tevzadze [38], which allows us to obtain existence and uniqueness of a solution when the terminal condition is small enough. Then, thanks to a well-chosen splitting, we recover an existence result for general bounded solution. Under additional assumptions, we can obtain stability results and a comparison theorem, which as usual implies uniqueness.