Uniqueness of solution to scalar BSDEs with $L\exp \left(\mu \sqrt{2\log (1+L)}\right)$-integrable terminal values

In [4], the existence of the solution is proved for a scalar linearly growing backward stochastic differential equation (BSDE) if the terminal value is $L\exp \left(\mu \sqrt{2\log (1+L)}\right)$-integrable with the positive parameter $\mu$ being bigger than a critical value $\mu \mathrm{\_}0$. In this note, we give the uniqueness result for the preceding BSDE.