Abstract
In this paper, we establish an analytic framework for studying set-valued backward stochastic differential equations (set-valued BSDE), motivated largely by the current studies of dynamic set-valued risk measures for multi-asset or network-based financial models. Our framework will make use of the notion of the Hukuhara difference between sets, in order to compensate the lack of “inverse” operation of the traditional Minkowski addition, whence the vector space structure in set-valued analysis. While proving the well-posedness of a class of set-valued BSDEs, we shall also address some fundamental issues regarding generalized Aumann–Itô integrals, especially when it is connected to the martingale representation theorem. In particular, we propose some necessary extensions of the integral that can be used to represent set-valued martingales with nonsingleton initial values. This extension turns out to be essential for the study of set-valued BSDEs.
Funding Statement
The first author was supported in part by Turkish NSF (TÜBİTAK) 3501-CAREER grant #117F438. This author acknowledges the additional support of the University of Southern California during a research visit for this work in January 2019.
The second author was supported in part by US NSF grant #2205972.
Acknowledgments
The authors would like to thank anonymous referees for their valuable suggestions to improve the paper, especially one who pointed out a serious issue in an earlier version of the paper.
Citation
Çağın Ararat. Jin Ma. Wenqian Wu. "Set-valued backward stochastic differential equations." Ann. Appl. Probab. 33 (5) 3418 - 3448, October 2023. https://doi.org/10.1214/22-AAP1896
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