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Existence and uniqueness of solution to scalar BSDEs with \(L\exp (\mu \sqrt{2\log (1+L)} )\)-integrable terminal values: the critical case. (English) Zbl 1422.60094
Summary: In [Y. Hu and S. Tang, Electron. Commun. Probab. 23, Paper No. 27, 11 p. (2018; Zbl 1390.60208)], 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 [R. Buckdahn et al., Electron. Commun. Probab. 23, Paper No. 59, 8 p. (2018; Zbl 1414.60040)] 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: \(\mu =\mu _0\).
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
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