[Paper Review] Path-Dependent Optimal Stochastic Control and Viscosity Solution of Associated Bellman Equations

Shanjian Tang, Fu Zhang|arXiv (Cornell University)|Oct 7, 2012

Stochastic processes and financial applications31 references3 citations

TL;DR

This paper establishes the viscosity solution theory for path-dependent fully nonlinear second-order PDEs arising from optimal stochastic control problems with path-dependent dynamics and recursive cost functionals. By introducing a novel viscosity solution concept restricted to α-Hölder continuous paths (α ∈ (0, 1/2)), and leveraging Dupire’s functional Itô calculus, the authors prove that the value function is the unique viscosity solution to the associated path-dependent Bellman equation.

ABSTRACT

In this paper we study the optimal stochastic control problem for a path-dependent stochastic system under a recursive path-dependent cost functional, whose associated Bellman equation from dynamic programming principle is a path-dependent fully nonlinear partial differential equation of second order. A novel notion of viscosity solutions is introduced. Using Dupire's functional Itô calculus, we characterize the value functional of the optimal stochastic control problem as the unique viscosity solution to the associated path-dependent Bellman equation.

Motivation & Objective

• To develop a viscosity solution theory for path-dependent fully nonlinear second-order PDEs arising in stochastic control.
• To address the lack of local compactness and non-smoothness in the path space by restricting semi-jets to α-Hölder continuous paths with α ∈ (0, 1/2).
• To characterize the value function of a path-dependent optimal stochastic control problem as the unique viscosity solution of the associated Bellman equation.
• To extend classical viscosity solution theory to the functional Itô calculus framework, overcoming limitations of Hilbert space and classical smoothness assumptions.
• To provide a rigorous foundation for solving non-Markovian optimal control problems with delay and path-dependent coefficients.

Proposed method

• Introduce a novel notion of viscosity solution by restricting semi-jets to the α-Hölder space C^α for α ∈ (0, 1/2), ensuring local compactness while managing non-smoothness.
• Apply Dupire’s functional Itô calculus to derive the path-dependent Bellman equation from the dynamic programming principle.
• Use the functional Itô formula to connect the value function to a path-dependent PDE, enabling the analysis of non-Markovian systems.
• Establish comparison and existence theorems for viscosity solutions in the functional setting via a perturbation argument and limiting procedure.
• Prove the value function is the unique viscosity solution by contradiction, using a carefully constructed test function and estimates on increments.
• Leverage α-Hölder continuity of the SDE solution (Proposition 7.1) to control pathwise regularity and support the viscosity solution framework.

Experimental results

Research questions

• RQ1 How can viscosity solutions be defined for path-dependent fully nonlinear PDEs when the path space lacks local compactness and the supremum norm is non-smooth?
• RQ2 Can the value function of a path-dependent optimal stochastic control problem be characterized as a viscosity solution to its associated Bellman equation?
• RQ3 What role does Dupire’s functional Itô calculus play in formulating and solving path-dependent Bellman equations?
• RQ4 How can the classical viscosity solution theory be adapted to infinite-dimensional path spaces with limited regularity?
• RQ5 What conditions ensure the uniqueness and existence of viscosity solutions in the context of path-dependent stochastic control?

Key findings

• The value function of the path-dependent optimal stochastic control problem is the unique viscosity solution to the associated path-dependent Bellman equation.
• A new viscosity solution concept is introduced by restricting semi-jets to the α-Hölder space C^α for α ∈ (0, 1/2), which resolves issues of non-compactness and non-smoothness.
• The proof of uniqueness relies on a contradiction argument using a perturbed test function and estimates on the increments of the value function.
• The authors establish α-Hölder continuity of the solution to path-dependent SDEs under uniform Lipschitz conditions, with the probability of large Hölder norms decaying as T^{(1/2−α)p}μ^{-p}.
• The framework successfully extends classical viscosity theory to non-Markovian settings, including delay SDEs and recursive path-dependent cost functionals.
• The result confirms that the dynamic programming principle leads to a well-defined path-dependent Bellman equation solvable in the viscosity sense.