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.

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