We address our interest to the development of a theory of viscosity solutions à la Crandall–Lions for path-dependent partial differential equations (PDEs), namely PDEs in the space of continuous paths . Path-dependent PDEs can play a central role in the study of certain classes of optimal control problems, as for instance optimal control problems with delay. Typically, they do not admit a smooth solution satisfying the corresponding HJB equation in a classical sense, it is therefore natural to search for a weaker notion of solution. While other notions of generalized solution have been proposed in the literature, the extension of the Crandall–Lions framework to the path-dependent setting is still an open problem. The question of uniqueness of the solutions, which is the most delicate issue, will be based on early ideas from the theory of viscosity solutions and a suitable variant of Ekeland’s variational principle. This latter is based on the construction of a smooth gauge-type function, where smooth is meant in the horizontal/vertical (rather than Fréchet) sense. In order to make the presentation more readable, we address the path-dependent heat equation, which in particular simplifies the smoothing of its natural “candidate” solution. Finally, concerning the existence part, we provide a functional Itô formula under general assumptions, extending earlier results in the literature.
The authors are very grateful to Prof. Mikhail Gomoyunov for his careful reading of the first version of the paper, for his comments and the very challenging questions. They also acknowledge the Referees, the Associate Editor and the Editor in Chief for their stimulating comments. The work of the second named author was supported by a public grant as part of the Investissement d’avenir project, reference ANR-11-LABX-0056-LMH, LabEx LMH, in a joint call with Gaspard Monge Program for optimization, operations research and their interactions with data sciences.
"Crandall–Lions viscosity solutions for path-dependent PDEs: The case of heat equation." Bernoulli 28 (1) 481 - 503, February 2022. https://doi.org/10.3150/21-BEJ1353