SET-VALUED BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS

• 成果类型: Article
• 署名作者: Ararat, Cagin; Ma, Jin; Wu, Wenqian
• 署名单位: Ihsan Dogramaci Bilkent University; University of Southern California
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/22-AAP1896
• 发表日期: 2023
• 页码: 3418-3448
• 关键词: integrals; expectations; consistent; convex

摘要

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-Ito 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.