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HAMILTON-JACOBI THEORY FOR A GENERALIZED
OPTIMAL STOPPING TIME PROBLEM

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J. J. Ye and Q. J. Zhu

**Abstract**: 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 (\mbox{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.

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