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

TL;DR

This paper proves existence, uniqueness, comparison, and stability theorems for unbounded solutions of scalar BSDEs with sub-quadratic growth in the generator, extending the nonlinear Feynman-Kac formula under weaker integrability conditions.

Contribution

It introduces new existence and comparison results for unbounded BSDE solutions with sub-quadratic generators under weaker integrability assumptions, and extends the nonlinear Feynman-Kac formula.

Findings

Existence of unbounded solutions under sub-exponential integrability.

Uniqueness and comparison theorems for convex/concave generators.

Stability results and extension to non-convex cases.

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 $|z|^\alpha$ for some $\alpha\in (1,2)$, when the terminal condition satisfies a sub-exponential moment integrability condition like $\exp\left(\mu L^{2/\alpha^*}\right)$ for the conjugate $\alpha^*$ of $\alpha$ and a positive parameter $\mu>\mu_0$ with a certain value $\mu_0$, which is clearly weaker than the usual $\exp(\mu L)$ 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.