OPTIMAL POSITION TARGETING VIA DECOUPLING FIELDS

• 成果类型: Article
• 署名作者: Ankirchner, Stefan; Fromm, Alexander; Kruse, Thomas; Popier, Alexandre
• 署名单位: Friedrich Schiller University of Jena; Justus Liebig University Giessen; Le Mans Universite
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/19-AAP1511
• 发表日期: 2020
• 页码: 644-672
• 关键词: STOCHASTIC DIFFERENTIAL-EQUATIONS; singular terminal condition; forward-backward sdes; Optimal transportation; BSDEs

摘要

We consider a variant of the basic problem of the calculus of variations, where the Lagrangian is convex and subject to randomness adapted to a Brownian filtration. We solve the problem by reducing it, via a limiting argument, to an unconstrained control problem that consists in finding an absolutely continuous process minimizing the expected sum of the Lagrangian and the deviation of the terminal state from a given target position. Using the Pontryagin maximum principle, we characterize a solution of the unconstrained control problem in terms of a fully coupled forward-backward stochastic differential equation (FBSDE). We use the method of decoupling fields for proving that the FBSDE has a unique solution. We exploit a mono-tonicity property of the decoupling field for solving the original constrained problem and characterize its solution in terms of an FBSDE with a free backward part.