Quadratic BSDEs with jumps: related non-linear expectations

TL;DR

This paper extends the theory of quadratic backward stochastic differential equations with jumps, establishing properties of associated non-linear expectations, including decomposition, regularity, and dual representations, with applications to dynamic risk measures.

Contribution

It introduces a comprehensive analysis of quadratic BSDEs with jumps, including existence, uniqueness, and properties of related non-linear expectations, along with dual representations and examples.

Findings

Non-linear Doob-Meyer decomposition for g-submartingales

Downcrossing inequality implies regularity in time

Converse comparison theorem for quadratic BSDEs with jumps

Abstract

In this article, we follow the study of quadratic backward SDEs with jumps,that is to say for which the generator has quadratic growth in the variables (z; u), started in our accompanying paper [15]. Relying on the existence and uniqueness result of [15], we define the corresponding g-expectations and study some of their properties. We obtain in particular a non-linear Doob-Meyer decomposition for g-submartingales and a downcrossing inequality which implies their regularity in time. As a consequence of these results, we also obtain a converse comparison theorem for our class of BSDEs. Finally, we provide a dual representation for the corresponding dynamic risk measures, and study the properties of their inf-convolution, giving several explicit examples