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Discussion of dynamic programming and linear programming approaches to stochastic control and optimal stopping in continuous time

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  • R. Stockbridge

Abstract

This paper seeks to highlight two approaches to the solution of stochastic control and optimal stopping problems in continuous time. Each approach transforms the stochastic problem into a deterministic problem. Dynamic programming is a well-established technique that obtains a partial/ordinary differential equation, variational or quasi-variational inequality depending on the type of problem; the solution provides the value of the problem as a function of the initial position (the value function). The other method recasts the problems as linear programs over a space of feasible measures. Both approaches use Dynkin’s formula in essential but different ways. The aim of this paper is to present the main ideas underlying these approaches with only passing attention paid to the important and necessary technical details. Copyright Springer-Verlag Berlin Heidelberg 2014

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  • R. Stockbridge, 2014. "Discussion of dynamic programming and linear programming approaches to stochastic control and optimal stopping in continuous time," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 77(1), pages 137-162, January.
    • Handle: RePEc:spr:metrik:v:77:y:2014:i:1:p:137-162
    DOI: 10.1007/s00184-013-0476-2
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    1. P. Kaczmarek & S. Kent & G. Rus & R. Stockbridge & B. Wade, 2007. "Numerical solution of a long-term average control problem for singular stochastic processes," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 66(3), pages 451-473, December.
    2. Broadie, Mark & Detemple, Jerome, 1995. "American Capped Call Options on Dividend-Paying Assets," The Review of Financial Studies, Society for Financial Studies, vol. 8(1), pages 161-191.
    3. L. Alili & A. E. Kyprianou, 2005. "Some remarks on first passage of Levy processes, the American put and pasting principles," Papers math/0508487, arXiv.org.
    4. Luis H. R. Alvarez, 2001. "Reward functionals, salvage values, and optimal stopping," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 54(2), pages 315-337, December.
    5. Dayanik, Savas & Karatzas, Ioannis, 2003. "On the optimal stopping problem for one-dimensional diffusions," Stochastic Processes and their Applications, Elsevier, vol. 107(2), pages 173-212, October.
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