Existence, uniqueness and comparison theorem on unbounded solutions of scalar super-linear BSDEs

TL;DR

This paper investigates the existence, uniqueness, and comparison theorems for unbounded solutions of scalar super-linear BSDEs with generators growing in a specific super-linear manner, under various integrability conditions based on the growth parameter.

Contribution

It establishes the weakest possible integrability conditions for solutions of super-linear BSDEs across different growth regimes and proves comparison theorems under convexity or continuity conditions.

Findings

Different integrability conditions depending on the growth parameter b4;

Existence of unbounded solutions under these conditions;

Comparison theorem and uniqueness results under convexity or Osgood conditions.

Abstract

This paper is devoted to the existence, uniqueness and comparison theorem on unbounded solutions of a scalar backward stochastic differential equation (BSDE) whose generator grows (with respect to both unknown variables $y$ and $z$) in a super-linear way like $|y||\ln |y||^{(\lambda+1/2)\wedge 1}+|z||\ln |z||^{\lambda}$ for some $\lambda\geq 0$. For the following four different ranges of the growth power parameter $\lambda$: $\lambda=0$, $\lambda\in (0,1/2)$, $\lambda=1/2$ and $\lambda>1/2$, we give reasonably weakest possible different integrability conditions of the terminal value for the existence of an unbounded solution to the BSDE. In the first two cases, they are stronger than the $L\ln L$-integrability and weaker than any $L^p$-integrability with $p>1$; in the third case, the integrability condition is just some $L^p$-integrability for $p>1$; and in the last case, the integrability condition is stronger than any $L^p$-integrability with $p>1$ and weaker than any $\exp(L^\epsilon)$-integrability with $\epsilon\in (0,1)$. We also establish the comparison theorem, which yields naturally the uniqueness, when either generator of both BSDEs is convex (concave) in both unknown variables $(y,z)$, or satisfies a one-sided Osgood condition in the first unknown variable $y$ and a uniform continuity condition in the second unknown variable $z$.