Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-25T06:17:47.588Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Part of: Markov processes Stochastic 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: [email protected]
∗∗ Postal address: Department of Mathematical Sciences, University of Wisconsin-Milwaukee, Milwaukee, WI 53201-0413, USA. Email address: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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.

References

Alili, L. and Kyprianou, A. E. (2005). Some remarks on first passage of Lévy processes, the American put and smooth pasting principles. Ann. Appl. Prob. 15, 2062–2080.CrossRefGoogle Scholar
Alvarez, L. H. R. (2001). Reward functionals, salvage values, and optimal stopping. Math. Meth. Operat. Res. 54, 315–337.CrossRefGoogle Scholar
Borodin, A. N. and Salminen, P. (2002). Handbook of Brownian Motion—Facts and Formulae, 2nd edn. Birkhäuser, Basel.Google Scholar
Dayanik, S. and Karatzas, I. (2003). On the optimal stopping problem for one-dimensional diffusions. Stoch. Process. Appl. 107, 173–212.CrossRefGoogle Scholar
Dynkin, E. B. (1965). Markov Processes, Vol. 2. Springer, New York.Google Scholar
El Karoui, N. (1981). Les aspects probabilistes du contrôle stochastique. In Lecture Notes in Math. 876, Springer, Berlin, pp. 73–238.Google Scholar
Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes. John Wiley, New York.Google Scholar
Helmes, K. and Stockbridge, R. H. (2007). Linear programming approach to the optimal stopping of singular stochastic processes. Stochastics 79, 309–335.CrossRefGoogle Scholar
Helmes, K. and Stockbridge, R. H. (2010). Thinning and harvesting in stochastic forest models. Preprint.Google Scholar
Itô, K. and McKean, H. P. Jr. (1974). Diffusion Process and Their Sample Paths. Springer, New York.Google Scholar
Øksendal, B. (2003). Stochastic Differential Equations, 6th edn. Springer, Berlin.CrossRefGoogle Scholar
Peskir, G. and Shiryaev, A. N. (2006). Optimal Stopping and Free-Boundary Problems. Birkhäuser, Basel.Google Scholar
Rogers, L. C. G. and Williams, D. (2000). Diffusions, Markov Processes and Martingales, Vol. 2. Cambridge University Press.Google Scholar
Salminen, P. (1985). Optimal stopping of one-dimensional diffusions. Math. Nachr. 124, 85–101.Google Scholar
Shiryaev, A. N. (1978). Optimal Stopping Rules. Springer, New York.Google Scholar
Zabczyk, J. (1979). Introduction to the theory of optimal stopping. In Stochastic Control Theory and Stochastic Differential Systems (Bad Honnef, 1979; Lecture Notes Control Inf. Sci. 16), Springer, Berlin, pp. 227–250.Google Scholar