BSE'S, BSDE'S AND FIXED-POINT PROBLEMS

• 成果类型: Article
• 署名作者: Cheridito, Patrick; Nam, Kihun
• 署名单位: Swiss Federal Institutes of Technology Domain; ETH Zurich; Rutgers University System; Rutgers University New Brunswick
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/16-AOP1149
• 发表日期: 2017
• 页码: 3795-3828
• 关键词: STOCHASTIC DIFFERENTIAL-EQUATIONS; convex generators; quadratic bsdes

摘要

In this paper, we introduce a class of backward stochastic equations (BSEs) that extend classical BSDEs and include many interesting examples of generalized BSDEs as well as semimartingale backward equations. We show that a BSE can be translated into a fixed-point problem in a space of random vectors. This makes it possible to employ general fixed-point arguments to establish the existence of a solution. For instance, Banach's contraction mapping theorem can be used to derive general existence and uniqueness results for equations with Lipschitz coefficients, whereas Schauder-type fixedpoint arguments can be applied to non-Lipschitz equations. The approach works equally well for multidimensional as for one-dimensional equations and leads to results in several interesting cases such as equations with path-dependent coefficients, anticipating equations, McKean-Vlasov-type equations and equations with coefficients of superlinear growth.