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CONSTRUCTION OF THE VALUE FUNCTION AND OPTIMAL RULES IN OPTIMAL STOPPING OF ONE-DIMENSIONAL DIFFUSIONS

KURT HELMES and RICHARD H. STOCKBRIDGE
Advances in Applied Probability
Vol. 42, No. 1 (MARCH 2010), pp. 158-182
Stable URL: http://www.jstor.org/stable/25683811
Page Count: 25
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Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
CONSTRUCTION OF THE VALUE FUNCTION AND OPTIMAL RULES IN OPTIMAL STOPPING OF ONE-DIMENSIONAL DIFFUSIONS
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Abstract

A new approach to the solution of optimal stopping problems for one-dimensional diffusions is developed. It arises by imbedding the stochastic problem in a linear programming problem over a space of measures. Optimizing over a smaller class of stopping rules provides a lower bound on the value of the original problem. Then the weak duality of a restricted form of the dual linear program provides an upper bound on the value. An explicit formula for the reward earned using a two-point hitting time stopping rule allows us to prove strong duality between these problems and, therefore, allows us to either optimize over these simpler stopping rules or to solve the restricted dual program. Each optimization problem is parameterized by the initial value of the diffusion and, thus, we are able to construct the value function by solving the family of optimization problems. This methodology requires little regularity of the terminal reward function. When the reward function is smooth, the optimal stopping locations are shown to satisfy the smooth pasting principle. The procedure is illustrated using two examples.

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