Uniqueness of solution to scalar BSDEs with $L\protect \qopname{}{o}{exp}\left(\mu _0\protect \sqrt{2\protect \qopname{}{o}{log}(1+L)}\right)$-integrable terminal values: an $L^1$-solution approach

Abstract

This paper deals with a class of scalar backward stochastic differential equations (BSDEs) with $L\exp (\mu _0\sqrt{2\log (1+L)})$-integrable terminal values for a critical parameter $\mu _0>0$. We show that the solution of these BSDEs is closely connected to the $L^1$-solution of the BSDEs with integrable parameters. The key tool is the Girsanov theorem. This idea leads to a new approach to the uniqueness of solution and we obtain a new existence and uniqueness result under general assumptions.