Stochatic Perron's method and verification without smoothness using viscosity comparison: the linear case

TL;DR

This paper develops a probabilistic Perron's method to construct viscosity solutions for linear parabolic equations linked to stochastic differential equations, enabling verification without smoothness assumptions.

Contribution

It introduces a probabilistic approach to Perron's method for viscosity solutions, facilitating verification in linear stochastic PDEs without requiring smoothness.

Findings

Constructed viscosity solutions using probabilistic Perron's method

Established a verification result equating viscosity solutions to expected payoffs

Provided a foundation for extending verification to stochastic control and game problems

Abstract

We introduce a probabilistic version of the classical Perron's method to construct viscosity solutions to linear parabolic equations associated to stochastic differential equations. Using this method, we construct easily two viscosity (sub and super) solutions that squeeze in between the expected payoff. If a comparison result holds true, then there exists a unique viscosity solution which is a martingale along the solutions of the stochastic differential equation. The unique viscosity solution is actually equal to the expected payoff. This amounts to a verification result (Ito's Lemma) for non-smooth viscosity solutions of the linear parabolic equation. This is the first step in a larger program to prove verification for viscosity solutions and the Dynamic Programming Principle for stochastic control problems and games