Stochastic Control and Exit Probabilities of Jump Processes

A stochastic control problem is formulated for some problems related to Markov process. This formulation is in some sense a generalization of one used in [2], [3], [4], [8] for diffusion case. We apply this to study the asymptotic behavior of exit probabilities of a family of jump processes depending on a small parameter ε as ε → 0.