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

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: $\mu ={\mu }_0$.