AN INFINITE-DIMENSIONAL APPROACH TO PATH-DEPENDENT KOLMOGOROV EQUATIONS

• 成果类型: Article
• 署名作者: Flandoli, Franco; Zanco, Giovanni
• 署名单位: University of Pisa
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/15-AOP1031
• 发表日期: 2016
• 页码: 2643-2693
• 关键词: space valued processes; VISCOSITY SOLUTIONS; covariation; pdes

摘要

In this paper, a Banach space framework is introduced in order to deal with finite-dimensional path-dependent stochastic differential equations. A version of Kolmogorov backward equation is formulated and solved both in the space of LP paths and in the space of continuous paths using the associated stochastic differential equation, thus establishing a relation between path-dependent SDEs and PDEs in analogy with the classical case. Finally, it is shown how to establish a connection between such Kolmogorov equation and the analogue finite-dimensional equation that can be formulated in terms of the path-dependent derivatives recently introduced by Dupire, Cont and Fournie.