# Existence and uniqueness of solution to scalar BSDEs with $L\exp\left(\mu\sqrt{2\log(1+L)}\right)$-integrable terminal values: the critical case

## TL;DR

This paper extends the existence and uniqueness results for scalar BSDEs to the critical integrability case where the parameter equals its threshold, filling a gap in the understanding of solutions under borderline conditions.

## Contribution

It proves that the existence and uniqueness of solutions for scalar BSDEs hold at the critical integrability threshold, completing previous results for subcritical cases.

## Key findings

- Existence of solutions at the critical integrability threshold.
- Uniqueness of solutions at the critical threshold.
- Extension of previous results to the boundary case.

## Abstract

In \cite{HuTang2018ECP}, the existence of the solution is proved for a scalar linearly growing backward stochastic differential equation (BSDE) when the terminal value is $L\exp\left(\mu\sqrt{2\log(1+L)}\right)$-integrable for a positive parameter $\mu>\mu_0$ with a critical value $\mu_0$, and a counterexample is provided to show that the preceding integrability for $\mu<\mu_0$ is not sufficient to guarantee the existence of the solution. Afterwards, the uniqueness result (with $\mu>\mu_0$) is also given in \cite{BuckdahnHuTang2018ECP} for the preceding BSDE under the uniformly Lipschitz condition of the generator. In this note, we prove that these two results still hold for the critical case: $\mu=\mu_0$.