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Construction of the Value Function and Optimal Rules in Optimal Stopping of One-Dimensional Diffusions

Part of: Stochastic processes Markov processes

Published online by Cambridge University Press:  01 July 2016

Kurt Helmes  and
Richard H. Stockbridge
Kurt Helmes*
Affiliation:
Humboldt-Universität zu Berlin
Richard H. Stockbridge*
Affiliation:
University of Wisconsin-Milwaukee
*
∗ Postal address: Institut für Operations Research, Humboldt-Universität zu Berlin, Germany. Email address: helmes@wiwi.hu-berlin.de
∗∗ Postal address: Department of Mathematical Sciences, University of Wisconsin-Milwaukee, Milwaukee, WI 53201-0413, USA. Email address: stockbri@uwm.edu
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Abstract

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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.

Keywords

Optimal stoppinglinear programmingdualitynonlinear optimizationvalue functionone-dimensional diffusion

MSC classification

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory
Secondary: 60J60: Diffusion processes
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 42 , Issue 1 , March 2010 , pp. 158 - 182
Copyright
Copyright © Applied Probability Trust 2010 

Footnotes

This research has been supported in part by the US National Security Agency under Grant Agreement Number H98230-09-1-0002. The United States Government is authorized to reproduce and distribute reprints notwithstanding any copyright notation herein.

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