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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 solutionifthe terminal value satisfies some $\Psi$-integrability condition, which isweaker 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 notsufficient for the existence of solutionto a BSDE of a linearly growinggenerator.

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