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

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