Stochatic Perron's method and verification without smoothness using viscosity comparison: the linear case
This work provides a foundational theoretical tool for verifying viscosity solutions in stochastic control and games, but it is limited to the linear case and is a first step in a larger program.
The authors introduce a probabilistic version of Perron's method to construct viscosity solutions for linear parabolic equations, showing that the unique viscosity solution equals the expected payoff, thereby providing a verification result for non-smooth viscosity solutions.
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