[Paper Review] Path-dependent Hamilton-Jacobi-Bellman equation: Uniqueness of Crandall-Lions viscosity solutions

Andrea Cosso, Fausto Gozzi+2 more|arXiv (Cornell University)|Jul 13, 2021

Stochastic processes and financial applications37 references4 citations

TL;DR

This paper establishes the uniqueness of Crandall-Lions viscosity solutions for path-dependent Hamilton-Jacobi-Bellman (HJB) equations arising in stochastic optimal control problems with non-Markovian dynamics. By leveraging approximation via finite-dimensional projections and regularity results for parabolic PDEs, the authors prove that the value function is the unique viscosity solution, resolving a key gap in the theory of path-dependent PDEs under general conditions.

ABSTRACT

We formulate a path-dependent stochastic optimal control problem under general conditions, for which weprove rigorously the dynamic programming principle and that the value function is the unique Crandall-Lions viscosity solution of the corresponding Hamilton-Jacobi-Bellman equation. Compared to the literature, the proof of our core result, that is the comparison theorem, is based on the fact that the valuefunction is bigger than any viscosity subsolution and smaller than any viscosity supersolution. It alsorelies on the approximation of the value function in terms of functions defined on finite-dimensionalspaces as well as on regularity results for parabolic partial differential equations.

Motivation & Objective

• To establish the uniqueness of Crandall-Lions viscosity solutions for path-dependent HJB equations in infinite-dimensional control problems.
• To rigorously prove the dynamic programming principle for path-dependent stochastic optimal control problems under general conditions.
• To bridge a gap in the literature by providing a comparison theorem for viscosity solutions in the path-dependent setting.
• To develop a framework where the value function is shown to dominate all subsolutions and be dominated by all supersolutions.
• To extend the viscosity solution theory to path-dependent PDEs using functional Itô calculus and cylindrical approximations.

Proposed method

• Formulates a path-dependent stochastic optimal control problem with non-anticipative coefficients and progressively measurable controls.
• Applies functional Itô calculus to define horizontal and vertical derivatives, enabling the formulation of path-dependent HJB equations.
• Uses cylindrical approximations to project the infinite-dimensional value function onto finite-dimensional spaces via regularized pathwise derivatives.
• Establishes regularity of approximated value functions using classical parabolic PDE theory and stability results.
• Employs a smooth variational principle and comparison theorem to show that the value function lies between any subsolution and supersolution.
• Proves convergence of approximated solutions to the true value function in the limit of increasing approximation dimension.

Experimental results

Research questions

• RQ1 Can the value function of a path-dependent stochastic control problem be uniquely characterized as a viscosity solution of the corresponding HJB equation?
• RQ2 Does the comparison principle hold for Crandall-Lions viscosity solutions in the context of path-dependent PDEs?
• RQ3 Can the value function be approximated by solutions of finite-dimensional PDEs while preserving viscosity solution properties?
• RQ4 How do the horizontal and vertical derivatives in functional Itô calculus facilitate the analysis of path-dependent HJB equations?
• RQ5 Under what conditions does the dynamic programming principle hold for non-Markovian SDEs with path-dependent coefficients?

Key findings

• The value function is the unique Crandall-Lions viscosity solution of the path-dependent HJB equation under general assumptions on the coefficients.
• The comparison theorem is established by showing that the value function is greater than any viscosity subsolution and less than any viscosity supersolution.
• The value function is approximated by a sequence of functions defined on finite-dimensional spaces, each solving a classical PDE with smooth coefficients.
• The approximated solutions converge pointwise to the true value function as the dimension of the approximation increases.
• The value function satisfies a Hölder-type continuity condition in time and path space, with a modulus depending on the problem's Lipschitz constants and time horizon.
• The viscosity solution property is preserved under the limit of cylindrical approximations, ensuring stability and uniqueness.