Stochastic Perron’s method and verification without smoothness using viscosity comparison: The linear case

We introduce a stochastic 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 (Itô’s Lemma) for non-smooth viscosity solutions of the linear parabolic equation.