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: $\mu=\mu_0$.
Categories: math.PR