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Parameter Dependent Optimal Thresholds, Indifference Levels and Inverse Optimal Stopping Problems

Part of:Stochastic processesMarkov processes

Published online by Cambridge University Press:  19 February 2016

Martin Klimmek
Martin Klimmek*
Affiliation:
University of Oxford
*
∗ Postal address: Nomura Centre for Mathematical Finance, Mathematical Institute, University of Oxford, 24-29 St Giles', Oxford OX1 3LB, UK. Email address: martinklimmek@gmail.com.
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Abstract

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Consider the classic infinite-horizon problem of stopping a one-dimensional diffusion to optimise between running and terminal rewards, and suppose that we are given a parametrised family of such problems. We provide a general theory of parameter dependence in infinite-horizon stopping problems for which threshold strategies are optimal. The crux of the approach is a supermodularity condition which guarantees that the family of problems is indexable by a set-valued map which we call the indifference map. This map is a natural generalisation of the allocation (Gittins) index, a classical quantity in the theory of dynamic allocation. Importantly, the notion of indexability leads to a framework for inverse optimal stopping problems.

Keywords

Inverse probleminverse optimal stoppingthreshold strategyparameter dependencecomparative staticsgeneralised diffusionGittins index

MSC classification

Primary:60G40: Stopping times; optimal stopping problems; gambling theory
Secondary:60J60: Diffusion processes

Information

Type
Research Article
Information
Journal of Applied Probability , Volume 51 , Issue 2 , June 2014, pp. 492 - 511
Copyright
© Applied Probability Trust 

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