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The moment sum of squares (moment-SOS) hierarchy produces sequences of upperand lower bounds on functionals of the exit time solution of a polynomialstochastic differential equation with polynomial constraints, at the price ofsolving semidefinite optimization problems of increasing size. In this note weuse standard results from elliptic partial differential equation analysis toprove convergence of the bounds produced by the hierarchy. We also useelementary convex analysis to describe a super- and sub-solution interpretationdual to a linear formulation on occupation measures. The practical relevance ofthe hierarchy is illustrated with numerical examples.

Moment-SOS hierarchy and exit time of stochastic processes