Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-17T12:19:49.666Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Part of: Markov processes Stochastic 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: [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.

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

References

Alfonsi, A. and Jourdain, B. (2008). General duality for perpetual American options. Internat. J. Theoret. Appl. Finance 11, 545–566.Google Scholar
Alvarez, L. H. R. (2001). Reward functionals, salvage values, and optimal stopping. Math. Meth. Operat. Res. 54, 315–337.Google Scholar
Athey, S. (1996). Comparative statics under uncertainty: Single crossing properties and log-supermodularity. Working paper, Department of Economics, MIT.Google Scholar
Bank, P. and Baumgarten, C. (2010). Parameter-dependent optimal stopping problems for one-dimensional diffusions. Electron. J. Prob. 15, 1971–1971.Google Scholar
Bensoussan, A. and Lions, J. L. (1982). Applications of variational inequalities in stochastic control (Stud. Math. Appl. 12). North-Holland Publishing Co., Amsterdam.Google Scholar
Black, F. (1988). Individual investment and consumption under uncertainty. In Portfolio Insurance: A Guide to Dynamic Hedging, ed. Luskin, D. L., 2nd edn. John Wiley, New York.Google Scholar
Borodin, A. N. and Salminen, P. (2002). Handbook of Brownian Motion – Facts and Formulae, 2nd edn. Birkhäuser, Basel.Google Scholar
Carlier, G. (2003). Duality and existence for a class of mass transportation problems and economic applications. Adv. Math. Econom. 5, 1–22.Google Scholar
Cox, A. M. G., Hobson, D. and Obloj, J. (2011). Utility theory front to back – inferring utility from agents' choices. Preprint arXiv:1101.3572.Google Scholar
Cox, J. C. and Leland, H. E. (2000). On dynamic investment strategies. J. Econom. Dynamics and Control 24, 1859–1880.Google Scholar
Dayanik, S. and Karatzas, I. (2003). On the optimal stopping problem for one-dimensional diffusions. Stoch. Process. Appl. 107, 173–212.Google Scholar
Ekström, E. and Hobson, D. (2011). Recovering a time-homogeneous stock price process from perpetual option prices. Ann. Appl. Prob. 21, 1102–1135.CrossRefGoogle Scholar
Gangbo, W. and McCann, R. J. (1996). The geometry of optimal transportation. Acta Math. 177, 113–161.Google Scholar
Gittins, J. C. and Glazebrook, K. D. (1977). On Bayesian models in stochastic scheduling. J. Appl. Prob. 14, 556–565.Google Scholar
Glazebrook, K. D., Hodge, D. J. and Kirkbride, C. (2011). General notions of indexability for queueing control and asset management. Ann. Appl. Prob. 21, 876–907.Google Scholar
He, H. and Huang, C. F. (1994). Consumption-portfolio policies: An inverse optimal problem. J. Econom. Theory 62, 257–293.Google Scholar
Hobson, D. and Klimmek, M. (2011). Constructing time-homogeneous generalized diffusions consistent with optimal stopping values. Stochastics 83, 477–503.Google Scholar
Jewitt, I. (1987). Risk aversion and the choice between risky prospects: The preservation of comparative statics results. Rev. Econom. Stud. 54, 73–85.Google Scholar
Karatzas, I. (1984). Gittins indices in the dynamic allocation problem for diffusion processes. Ann. Prob. 12, 173–192.Google Scholar
Klimmek, R. (1986). Risikoneigung und Besteuerung, University for business administration, Florentz, Munich.Google Scholar
Lu, B. (2010). Recovering a piecewise constant volatility from perpetual put option prices. J. Appl. Prob. 47, 680–692.Google Scholar
Milgrom, P. and Segal, I. (2002). Envelope theorems for arbitrary choice sets. Econometrica 70, 583–601.Google Scholar
Rogers, L. C. G. and Williams, D. (2000). Diffusions, Markov Processes and Martingales, Vol. 2. Cambridge University Press.Google Scholar
Rüschendorf, L. (1991). Fréchet-bounds and their applications. In Advances in Probability Distributions with Given Marginals: beyond the copulas, ed. Dall'Aglio, G. Kluwer Academic Publishers, Dordrecht.Google Scholar
Salminen, P. (1985). Optimal stopping of one-dimensional diffusions. Math. Nachr. 124, 85–101.Google Scholar
Samuelson, P. (1948). Consumption theory in terms of revealed preference. Economica 15, 243–253.Google Scholar
Samuelson, P. (1964). Tax deductibility of economic depreciation to insure invariant valuations. J. Political Econom. 72, 604–606.Google Scholar
Whittle, P. (1988). Restless bandits: Activity allocation in a changing world. J. Appl. Prob. 25A, 287–298.Google Scholar