## Electronic Communications in Probability

### Existence and uniqueness of solution to scalar BSDEs with $L\exp (\mu \sqrt{2\log (1+L)} )$-integrable terminal values: the critical case

#### 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: $\mu =\mu _{0}$.

#### Article information

Source
Electron. Commun. Probab., Volume 24 (2019), paper no. 49, 10 pp.

Dates
Accepted: 2 July 2019
First available in Project Euclid: 12 September 2019

https://projecteuclid.org/euclid.ecp/1568253711

Digital Object Identifier
doi:10.1214/19-ECP254

#### Citation

Fan, Shengjun; Hu, Ying. 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 (2019), paper no. 49, 10 pp. doi:10.1214/19-ECP254. https://projecteuclid.org/euclid.ecp/1568253711

#### References

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