Uniqueness of solution to scalar BSDEs with $L\exp{\left(\mu \sqrt{2\log{(1+L)}}\,\right)}$-integrable terminal values

TL;DR

This paper establishes the uniqueness of solutions for scalar BSDEs with terminal values that are integrable under a specific exponential-logarithmic condition, extending the understanding of solution properties under such integrability constraints.

Contribution

It provides the first proof of uniqueness for scalar BSDEs with terminal values having $L ext{exp}( ext{parameter} imes ext{logarithmic function})$-integrability, complementing existing existence results.

Findings

Proves uniqueness of solutions for scalar BSDEs with specified terminal value integrability.

Extends the theory of BSDEs to include solutions with exponential-logarithmic integrability conditions.

Abstract

In [4], the existence of the solution is proved for a scalar linearly growing backward stochastic differential equation (BSDE) if the terminal value is $L\exp{\left(\mu \sqrt{2\log{(1+L)}}\,\right)}$-integrable with the positive parameter $\mu$ being bigger than a critical value $\mu\_0$. In this note, we give the uniqueness result for the preceding BSDE.