FEYNMAN-KAC REPRESENTATION FOR HAMILTON-JACOBI- BELLMAN IPDE

• 成果类型: Article
• 署名作者: Kharroubi, Idris; Huyen Pham
• 署名单位: Universite PSL; Universite Paris-Dauphine; Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Centre National de la Recherche Scientifique (CNRS); Universite Paris Cite; Institut Polytechnique de Paris; ENSAE Paris
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/14-AOP920
• 发表日期: 2015
• 页码: 1823-1865
• 关键词: STOCHASTIC DIFFERENTIAL-EQUATIONS; jumps

摘要

We aim to provide a Feynman-Kac type representation for Hamilton Jacobi-Bellman equation, in terms of forward backward stochastic differential equation (FBSDE) with a simulatable forward process. For this purpose, we introduce a class of BSDE where the jumps component of the solution is subject to a partial nonpositive constraint. Existence and approximation of a unique minimal solution is proved by a penalization method under mild assumptions. We then show how minimal solution to this BSDE class provides a new probabilistic representation for nonlinear integro-partial differential equations (IPDEs) of Hamilton-Jacobi-Bellman (HJB) type, when considering a regime switching forward SDE in a Markovian framework, and importantly we do not make any ellipticity condition. Moreover, we state a dual formula of this BSDE minimal solution involving equivalent change of probability measures. This gives in particular an original representation for value functions of stochastic control problems including controlled diffusion coefficient.