BACKWARD STOCHASTIC DIFFERENTIAL EQUATION DRIVEN BY A MARKED POINT PROCESS: AN ELEMENTARY APPROACH WITH AN APPLICATION TO OPTIMAL CONTROL

• 成果类型: Article
• 署名作者: Confortola, Fulvia; Fuhrman, Marco; Jacod, Jean
• 署名单位: Polytechnic University of Milan; Sorbonne Universite; Universite Paris Cite; Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI)
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/15-AAP1132
• 发表日期: 2016
• 页码: 1743-1773
• 关键词: Jumps

摘要

We address a class of backward stochastic differential equations on a bounded interval, where the driving noise is a marked, or multivariate, point process. Assuming that the jump times are totally inaccessible and a technical condition holds (see Assumption (A) below), we prove existence and uniqueness results under Lipschitz conditions on the coefficients. Some counter-examples show that our assumptions are indeed needed. We use a novel approach that allows reduction to a (finite or infinite) system of deterministic differential equations, thus avoiding the use of martingale representation theorems and allowing potential use of standard numerical methods. Finally, we apply the main results to solve an optimal control problem for a marked point process, formulated in a classical way.