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

Abstract

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.

Keywords

Existence and uniqueness / Unbounded solutions / Backward stochastic differential equation / Comparison theorem / General time-interval / Sub-quadratic growth

Acknowledgements

This work is supported by the National Natural Science Foundation of China (Grant No. 12171471), the Natural Science Foundation of Jiangsu Province (Grant No. BK20231057), and the Graduate Innovation Program of China University of Mining and Technology (Grant No. 2023WLJCRCZL143). The authors would like to thank the anonymous referees for their careful reading and valuable suggestions.

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