On the uniqueness of solutions to quadratic BSDEs with convex generators and unbounded terminal conditions: the critical case

TL;DR

This paper establishes the uniqueness of solutions to quadratic backward stochastic differential equations with convex generators in the critical case where the exponential integrability matches a key threshold, under strong convexity assumptions.

Contribution

It extends the uniqueness results to the critical integrability case by proving uniqueness under strong convexity of the generator.

Findings

Uniqueness holds for solutions with exponential integrability in the critical case p=γ.

Strong convexity of the generator is essential for the uniqueness result.

The result bridges the gap between subcritical and critical cases in quadratic BSDEs.

Abstract

In [3], the authors proved that uniqueness holds among solutions whose exponentials are $L^p$ with $p$ bigger than a constant $\gamma$ ($p\textgreater{}\gamma$). In this paper, we consider the critical case: $p=\gamma$. We prove that the uniqueness holds among solutions whose exponentials are $L^\gamma$ under the additional assumption that the generator is strongly convex.