Weighted L^p~(p≥1) solutions of random time horizon BSDEs with stochastic monotonicity generators

2024/10/02 by Li, Xinying, Fan, Shengjun
#FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2410.01543

Abstract

In this paper, we are concerned with a multidimensional backward stochastic differential equation (BSDE) with a general random terminal time τ, which may take values in [0,+∞]. Firstly, we establish an existence and uniqueness result for a weighted L^p~(p>1) solution of the preceding BSDE with generator g satisfying a stochastic monotonicity condition with general growth in the first unknown variable y and a stochastic Lipschitz continuity condition in the second unknown variable z. Then, we derive an existence and uniqueness result for a weighted L¹ solution of the preceding BSDE under an additional stochastic sub-linear growth condition in z. These results generalize the corresponding ones obtained in \citeLi2024 to the L^p~(p≥ 1) solution case. Finally, the corresponding comparison theorems for the weighted L^p~(p≥1) solutions are also put forward and verified in the one-dimensional setting. In particular, we develop new ideas and systematical techniques in order to establish the above results.

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