General time interval multidimensional BSDEs with generators satisfying a weak stochastic-monotonicity condition

Tingting Li; Ziheng Xu; Shengjun Fan

doi:10.3934/puqr.2021015

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2021, Volume 6, Issue 4: 301-318. Doi: 10.3934/puqr.2021015

This issue Previous Article Conditional coherent risk measures and regime-switching conic pricing Next Article Existence, uniqueness and strict comparison theorems for BSDEs driven by RCLL martingales

General time interval multidimensional BSDEs with generators satisfying a weak stochastic-monotonicity condition

1.
School of Mathematics, China University of Mining and Technology, Xuzhou 221116, Jiangsu, China

f_s_j@126.com (Shengjun Fan)
f_s_j@126.com (Shengjun Fan)

Received: July 04, 2021

Accepted: October 27, 2021

Published: December 2021

The authors are greatly grateful to the referee for his/her careful reading and valuable suggestions.Fan’s research is partially supported by the Fundamental Research Funds for the CentralUniversities (Grant No. 2017XKZD11) and the National Natural Science Foundation of China(Grant No. 12171471).

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Abstract

This paper establishes an existence and uniqueness result for the adapted solution of a general time interval multidimensional backward stochastic differential equation (BSDE), where the generator $ g $ satisfies a weak stochastic-monotonicity condition and a general growth condition in the state variable $ y $, and a stochastic-Lipschitz condition in the state variable $ z $. This unifies and strengthens some known works. In order to prove this result, we develop some ideas and techniques employed in Xiao and Fan [25] and Liu et al. [15]. In particular, we put forward and prove a stochastic Gronwall-type inequality and a stochastic Bihari-type inequality, which generalize the classical ones and may be useful in other applications. The martingale representation theorem, Itô’s formula, and the BMO martingale tool are used to prove these two inequalities.

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Mathematics Subject Classification: 60H10.

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References

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