Existence of solution to scalar BSDEs with weakly $L^{1+}$-integrable terminal values

TL;DR

This paper investigates the existence of solutions for scalar BSDEs with terminal values that are weakly integrable, introducing a new integrability condition that is weaker than traditional $L^p$ but stronger than $L ext{log}L$.

Contribution

It establishes a new $ ext{ extPhi}$-integrability condition ensuring solutions for scalar BSDEs with weakly $L^{1+}$-integrable terminal values, extending previous integrability requirements.

Findings

Solutions exist under the new $ ext{ extPhi}$-integrability condition.

$L ext{log}L$ integrability alone is insufficient for existence.

Counterexample demonstrates the necessity of the new condition.

Abstract

In this paper, we study a scalar linearly growing BSDE with a weakly $L^{1+}$-integrable terminal value. We prove that the BSDE admits a solutionif the terminal value satisfies some $\Psi$-integrability condition, which is weaker than the usual $L^p$ ($p>1$) integrability and stronger than $L\log L$ integrability. We show by a counterexample that $L\log L$ integrability is not sufficient for the existence of solutionto a BSDE of a linearly growing generator.