Existence, uniqueness, comparison theorem and stability theorem for unbounded solutions of scalar BSDEs with sub-quadratic generators

Abstract

We first establish the existence of an unbounded solution to a backward stochastic differential equation (BSDE) with generator $g$ allowing a general growth in the state variable $y$ and a sub-quadratic growth in the state variable $z$, like $|z{|}^{\hat{I}}\pm$ for some $\hat{I}\pm \in (1,2)$, when the terminal condition satisfies a sub-exponential moment integrability condition like $\exp \left(\hat{I}¼L^{2/\hat{I}{\pm }^{\ast }}\right)$ for the conjugate $\hat{I}{\pm }^{\ast }$ of $\hat{I}\pm$ and a positive parameter $\hat{I}¼>\hat{I}{¼}_0$ with a certain value $\hat{I}{¼}_0$, which is clearly weaker than the usual $\exp (\hat{I}¼L)$ integrability and stronger than $L^p(p>1)$ integrability. Then, we prove the uniqueness and comparison theorem for the unbounded solutions of the preceding BSDEs under the additional assumptions that the terminal conditions have sub-exponential moments of any order and the generators are convex or concave in $(y,z)$. Afterwards, we extend the uniqueness and comparison theorem to the non-convexity and non-concavity case, and establish a general stability result for the unbounded solutions of the preceding BSDEs. Finally, with these tools in hands, we derive the nonlinear Feynman-Kac formula in this context.

32 pages