PATH-DEPENDENT EQUATIONS AND VISCOSITY SOLUTIONS IN INFINITE DIMENSION

• 成果类型: Article
• 署名作者: Cosso, Andrea; Federico, Salvatore; Gozzi, Fausto; Rosestolato, Mauro; Touzi, Nizar
• 署名单位: Polytechnic University of Milan; University of Siena; Luiss Guido Carli University; Luiss Guido Carli University; Institut Polytechnique de Paris; Ecole Polytechnique
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/17-AOP1181
• 发表日期: 2018
• 页码: 126-174
• 关键词: optimal stochastic-control; nonlinear 2nd-order equations; differential-equations; Bellman equations; SPACES

摘要

Path-dependent partial differential equations (PPDEs) are natural objects to study when one deals with non-Markovian models. Recently, after the introduction of the so-called pathwise (or functional or Dupire) calculus [see Dupire (2009)], in the case of finite-dimensional underlying space various papers have been devoted to studying the well-posedness of such kind of equations, both from the point of view of regular solutions [see, e.g., Dupire (2009) and Cont (2016) Stochastic Integration by Parts and Functional Ito Calculus 115-207, Birkhauser] and viscosity solutions [see, e.g., Ekren et al. (2014) Ann. Probab. 42 204-236]. In this paper, motivated by the study of models driven by path-dependent stochastic PDEs, we give a first well-posedness result for viscosity solutions of PPDEs when the underlying space is a separable Hilbert space. We also observe that, in contrast with the finite-dimensional case, our well-posedness result, even in the Markovian case, applies to equations which cannot be treated, up to now, with the known theory of viscosity solutions.