  Among various rare events, the effective computation of transition paths
connecting metastable states in a stochastic model is an important problem.
This paper proposes a stochastic optimal control formulation for transition
path problems in an infinite time horizon for Markov jump processes on polish
space. An unbounded terminal cost at a stopping time and a controlled
transition rate for the jump process regulate the transition from one
metastable state to another. The running cost is taken as an entropy form of
the control velocity, in contrast to the quadratic form for diffusion
processes. Using the Girsanov transformation for Markov jump processes, the
optimal control problem in both finite time and infinite time horizon with
stopping time fit into one framework: the optimal change of measures in the
C\`adl\`ag path space via minimizing their relative entropy. We prove that the
committor function, solved from the backward equation with appropriate boundary
conditions, yields an explicit formula for the optimal path measure and the
associated optimal control for the transition path problem. The unbounded
terminal cost leads to a singular transition rate (unbounded control velocity),
for which, the Gamma convergence technique is applied to pass the limit for a
regularized optimal path measure. The limiting path measure is proved to solve
a Martingale problem with an optimally controlled transition rate and the
associated optimal control is given by Doob-h transformation. The resulting
optimally controlled process can realize the transitions almost surely.
