Existence, uniqueness and comparison theorem on unbounded solutions of general time-interval BSDEs with sub-quadratic generators

This study addresses the existence, uniqueness, and comparison theorem for unbounded solutions of one-dimensional backward stochastic differential equations (BSDEs) with sub-quadratic generators, considering both finite and infinite terminal times. Initially, we establish the existence of unbounded solutions for BSDEs where the generator $ g $ satisfies a time-varying one-sided linear growth condition in the first unknown variable $ y $ and a time-varying sub-quadratic growth condition in the second unknown variable $ z $. Next, the uniqueness and comparison theorems for unbounded solutions are proven under a time-varying extended convexity assumption. These findings extend the results in [12] to the general time-interval BSDEs. Finally, we propose and verify several sufficient conditions for ensuring uniqueness, utilizing innovative approaches applied for the first time, even in the context of finite time-interval BSDEs.