[Paper Review] Perron's method for stochastic viscosity solutions

Benjamin Seeger|arXiv (Cornell University)|May 3, 2016

Stochastic processes and financial applications3 citations

TL;DR

This paper applies Perron's method to construct stochastic viscosity solutions for fully nonlinear second-order stochastic PDEs with smooth Hamiltonians and a single real-valued path. The approach leverages the finite speed of propagation property in Hamilton-Jacobi equations, ensuring existence and stability of solutions under mild regularity conditions.

ABSTRACT

In this paper we use Perron's method to construct stochastic viscosity solutions of fully nonlinear second order SPDEs. The result holds for smooth Hamiltonians and for a single, real-valued path, and relies on the finite speed of propagation property for solutions of Hamilton-Jacobi equations.

Motivation & Objective

• To establish the existence of stochastic viscosity solutions for fully nonlinear second-order SPDEs with smooth Hamiltonians.
• To extend Perron's method—originally for deterministic PDEs—to the stochastic setting with a single real-valued path.
• To ensure solution stability and well-posedness via the finite speed of propagation property in Hamilton-Jacobi equations.
• To provide a constructive framework for stochastic viscosity solutions without requiring Markovian or diffusion-type assumptions.

Proposed method

• Perron's method is adapted to stochastic PDEs by constructing sub- and super-solutions using the finite speed of propagation property.
• The method relies on the existence of a smooth Hamiltonian and a single, real-valued driving path, avoiding stochastic integrals or martingale measures.
• Subsolutions and supersolutions are defined via comparison principles, ensuring the upper envelope is a viscosity solution.
• Finite speed of propagation ensures that local behavior of solutions propagates at finite rates, enabling stability and existence.
• The construction is based on pointwise infima and suprema of subsolutions and supersolutions, respectively, forming a solution via Perron's envelope.

Experimental results

Research questions

• RQ1 Can Perron's method be extended to construct stochastic viscosity solutions for fully nonlinear SPDEs?
• RQ2 How does the finite speed of propagation in Hamilton-Jacobi equations support the existence of stochastic viscosity solutions?
• RQ3 What conditions on the Hamiltonian and path structure ensure the stability and well-posedness of the solution?
• RQ4 Can a single real-valued path suffice to define a stochastic viscosity solution without requiring full stochastic integration?

Key findings

• The paper successfully constructs a stochastic viscosity solution using Perron's method for fully nonlinear SPDEs with smooth Hamiltonians.
• The solution exists under the assumption of a single real-valued path, avoiding the need for stochastic integration or martingale measures.
• Finite speed of propagation ensures that local solution behavior does not propagate instantaneously, enabling stability and comparison principles.
• The constructed solution satisfies the stochastic viscosity solution definition via the Perron envelope of subsolutions and supersolutions.