  In this paper, we derive a parabolic partial differential equation for the
expected exit time of non-autonomous time-periodic non-degenerate stochastic
differential equations. This establishes a Feynman-Kac duality between expected
exit time of time-periodic stochastic differential equations and time-periodic
solutions of parabolic partial differential equations. Casting the
time-periodic solution of the parabolic partial differential equation as a
fixed point problem and a convex optimisationproblem, we give sufficient
conditions in which the partial differential equation is well-posed in a weak
and 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), the
method in this paper is applicable to wide range of physical systems including
weakly dissipative systems. Particular applications towards stochastic
resonance will be discussed.
