[Paper Review] Stochastic Representations for Solutions to Parabolic Dirichlet Problems for Nonlocal Bellman Equations

Ruoting Gong, Chenchen Mou+1 more|arXiv (Cornell University)|Sep 1, 2017

Stochastic processes and financial applications4 citations

TL;DR

This paper establishes a stochastic representation formula for the viscosity solution of a degenerate Hamilton-Jacobi-Bellman (HJB) integro-partial differential equation (integro-PDE) in a bounded domain, proving that the unique viscosity solution equals the value function of an associated stochastic optimal control problem. The proof relies on approximation techniques, viscosity solution theory, and careful analysis of exit times and Lévy processes, with the dynamic programming principle derived as a key result.

ABSTRACT

We prove a stochastic representation formula for the viscosity solution of Dirichlet terminal-boundary value problem for a degenerate Hamilton-Jacobi-Bellman integro-partial differential equation in a bounded domain. We show that the unique viscosity solution is the value function of the associated stochastic optimal control problem. We also obtain the dynamic programming principle for the associated stochastic optimal control problem in a bounded domain.

Motivation & Objective

• To establish a stochastic representation formula for the viscosity solution of a degenerate HJB integro-PDE with Dirichlet terminal-boundary conditions in a bounded domain.
• To prove that the unique viscosity solution coincides with the value function of an associated stochastic optimal control problem.
• To rigorously derive the dynamic programming principle for the stochastic control problem in a bounded domain, addressing technical gaps in prior literature.
• To handle the challenges of bounded domains through approximation methods involving non-degenerate, finite-control, and smooth-approximated equations.

Proposed method

• Approximate the original degenerate HJB integro-PDE with non-degenerate equations on slightly enlarged domains using sequences of coefficients with finite control sets and smooth data.
• Construct viscosity supersolutions and subsolutions via barrier functions and cutoff techniques, ensuring uniform continuity and boundary consistency.
• Use Perron’s method to obtain the viscosity solution of the original equation, then verify it satisfies the value function property via approximation limits.
• Apply the theory of viscosity solutions for integro-PDEs from [31], relying on mild regularity assumptions on the domain and non-degeneracy along the boundary.
• Analyze exit times and behavior of Lévy-driven diffusion processes to control boundary effects and ensure convergence of approximations.
• Establish the dynamic programming principle by verifying the verification condition through uniform bounds and viscosity solution stability.

Experimental results

Research questions

• RQ1 Can the unique viscosity solution of a degenerate HJB integro-PDE in a bounded domain be represented as the value function of a stochastic optimal control problem?
• RQ2 Does the dynamic programming principle hold for stochastic control problems driven by Lévy processes in bounded domains with Dirichlet boundary conditions?
• RQ3 How can one construct uniform viscosity supersolutions and subsolutions that respect the boundary data and terminal condition in the presence of nonlocal operators?
• RQ4 What approximation scheme ensures convergence of classical solutions to the viscosity solution of the original problem in a bounded domain?
• RQ5 Under what regularity and non-degeneracy assumptions on the domain and coefficients does the value function remain continuous and satisfy the HJB integro-PDE?

Key findings

• The unique viscosity solution of the Dirichlet terminal-boundary value problem for the degenerate HJB integro-PDE is identified as the value function of the associated stochastic optimal control problem.
• A stochastic representation formula is established, linking the solution to the expected cost-to-go of controlled jump-diffusion processes in bounded domains.
• The dynamic programming principle is rigorously proven for the stochastic control problem, extending known results to bounded domains with nonlocal dynamics.
• Uniform continuity of the constructed viscosity subsolutions and supersolutions is achieved, independent of approximation parameters, ensuring stability in the limit.
• The proof relies on a novel approximation scheme using non-degenerate, finite-control, and smooth-approximated equations, with convergence established via precise estimates on exit times and process behavior.
• The construction of barrier functions with uniform lower bounds on gradients ensures the validity of viscosity supersolution properties across the domain boundary.