Optimal Stopping with a Horizon Constraint

A Langrangean function approach to optimality and duality is used to study the optimal stopping problem with a horizon constraint. This approach forces the analysis of the problem to “look beyond” the horizon time, which is essential to any implicit valuation of the horizon constraint. The results of this approach include the usual relationships between solutions to the primal and dual problems, optimality conditions, and saddlepoints of the Lagrangean function, as well as necessary optimality and strong duality theorems under a stability hypothesis. The constrained stopping problem is analogous in its simplicity to classical single equation constrained optimization in that an optimal multiplier is easily identified.