On a problem of stochastic reach-avoid set characterization

We develop a novel framework for formulating a class of stochastic reachability problems with state constraints as a stochastic optimal control problem. Previous approaches to solving these problems are either confined to the deterministic setting or address almost-sure stochastic notions. In contrast, we propose a new methodology to tackle probabilistic specifications that are less specific than almost sure requirements. To this end, we first establish a connection between two stochastic reach-avoid problems and a class of stochastic optimal control problems for diffusions with discontinuous payoff functions. We then derive a weak version of dynamic programming principle (DPP) for the value function. Moreover, based on our DPP, we give an alternate characterization of the value function as the solution to a partial differential equation in the sense of discontinuous viscosity solutions. Finally we validate the performance of the proposed framework on the stochastic Zermelo navigation problem.