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In this article we approach a class of stochastic reachability problems withstate constraints from an optimal control perspective. Preceding approaches tosolving these reachability problems are either confined to the deterministicsetting or address almost-sure stochastic requirements. In contrast, we proposea methodology to tackle problems with less stringent requirements than almostsure. To this end, we first establish a connection between two distinctstochastic reach-avoid problems and three classes of stochastic optimal controlproblems involving discontinuous payoff functions. Subsequently, we focus onsolutions of one of the classes of stochastic optimal control problems---theexit-time problem, which solves both the two reach-avoid problems mentionedabove. We then derive a weak version of a dynamic programming principle (DPP)for the corresponding value function; in this direction our contributioncompared to the existing literature is to develop techniques that admitdiscontinuous payoff functions. Moreover, based on our DPP, we provide analternative characterization of the value function as a solution of a partialdifferential equation in the sense of discontinuous viscosity solutions, alongwith boundary conditions both in Dirichlet and viscosity senses. Theoreticaljustifications are also discussed to pave the way for deployment ofoff-the-shelf PDE solvers for numerical computations. Finally, we validate theperformance of the proposed framework on the stochastic Zermelo navigationproblem.

The Stochastic Reach-Avoid Problem and Set Characterization for Diffusions