Hamilton-Jacobi Theory For A Generalized Optimal Stopping Time Problem

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We study an optimal stopping time problem with an extended-valued lower semicontinuous stopping cost. We prove that the optimal value associated with such a problem as a function of the initial time and state is the unique lower semicontinuous solution of a generalized Hamilton-Jacobi equation (H-J equation for short) involving the proximal subgradients. We also derive a verification theorem that uses lower semicontinuous verification functions. Allowing the stopping cost to be extended-valued considerably generalizes the ordinary stopping time problem. The generalized optimal stopping time problem considered in this paper encompasses stopping time problems, fixed time problems, free time problems, infinite horizon problems and some types of exit time problems as special cases.