  We first derive the Hamilton-Jacobi theory underlying continuous-time Markov
processes, and then use the construction to develop a variational algorithm for
estimating escape (least improbable or first passage) paths for a generic
stochastic chemical reaction network that exhibits multiple fixed points. The
design of our algorithm is such that it is independent of the underlying
dimensionality of the system, the discretization control parameters are updated
towards the continuum limit, and there is an easy-to-calculate measure for the
correctness of its solution. We consider several applications of the algorithm
and verify them against computationally expensive means such as the shooting
method and stochastic simulation. While we employ theoretical techniques from
mathematical physics, numerical optimization and chemical reaction network
theory, we hope that our work finds practical applications with an
inter-disciplinary audience including chemists, biologists, optimal control
theorists and game theorists.
