Abstract
In [8], the existence of the solution is proved for a scalar linearly growingbackward stochastic differential equation (BSDE) when the terminal value is$L\exp (\mu \sqrt{2\log (1+L)} )$-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 [3] 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: $\mu =\mu _{0}$.
Citation
Shengjun Fan. Ying Hu. "Existence and uniqueness of solution to scalar BSDEs with $L\exp (\mu \sqrt{2\log (1+L)} )$-integrable terminal values: the critical case." Electron. Commun. Probab. 24 1 - 10, 2019. https://doi.org/10.1214/19-ECP254
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