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In this paper, we derive a parabolic partial differential equation for theexpected exit time of non-autonomous time-periodic non-degenerate stochasticdifferential equations. This establishes a Feynman-Kac duality between expectedexit time of time-periodic stochastic differential equations and time-periodicsolutions of parabolic partial differential equations. Casting thetime-periodic solution of the parabolic partial differential equation as afixed point problem and a convex optimisationproblem, we give sufficientconditions in which the partial differential equation is well-posed in a weakand classical sense. With no known closed formulae for the expected exit time,we show our method can be readily implemented by standard numerical schemes.With relatively weak conditions (e.g. locally Lipschitz coefficients), themethod in this paper is applicable to wide range of physical systems includingweakly dissipative systems. Particular applications towards stochasticresonance will be discussed.

Expected Exit Time for Time-Periodic Stochastic Differential Equations and Applications to Stochastic Resonance