Journal of Applied Probability

Parameter dependent optimal thresholds, indifference levels and inverse optimal stopping problems

Martin Klimmek

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Abstract

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.

Article information

Source
J. Appl. Probab., Volume 51, Number 2 (2014), 492-511.

Dates
First available in Project Euclid: 12 June 2014

Permanent link to this document
https://projecteuclid.org/euclid.jap/1402578639

Digital Object Identifier
doi:10.1239/jap/1402578639

Mathematical Reviews number (MathSciNet)
MR3217781

Zentralblatt MATH identifier
1327.60099

Subjects
Primary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60]
Secondary: 60J60: Diffusion processes [See also 58J65]

Keywords
Inverse problem inverse optimal stopping threshold strategy parameter dependence comparative statics generalised diffusion Gittins index

Citation

Klimmek, Martin. Parameter dependent optimal thresholds, indifference levels and inverse optimal stopping problems. J. Appl. Probab. 51 (2014), no. 2, 492--511. doi:10.1239/jap/1402578639. https://projecteuclid.org/euclid.jap/1402578639


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References

  • Alfonsi, A. and Jourdain, B. (2008). General duality for perpetual American options. Internat. J. Theoret. Appl. Finance 11, 545–566.
  • Alvarez, L. H. R. (2001). Reward functionals, salvage values, and optimal stopping. Math. Meth. Operat. Res. 54, 315–337.
  • Athey, S. (1996). Comparative statics under uncertainty: Single crossing properties and log-supermodularity. Working paper, Department of Economics, MIT.
  • Bank, P. and Baumgarten, C. (2010). Parameter-dependent optimal stopping problems for one-dimensional diffusions. Electron. J. Prob. 15, 1971–1993.
  • Bensoussan, A. and Lions, J. L. (1982). Applications of variational inequalities in stochastic control (Stud. Math. Appl. 12). North-Holland Publishing Co., Amsterdam.
  • Black, F. (1988). Individual investment and consumption under uncertainty. In Portfolio Insurance: A Guide to Dynamic Hedging, ed. D. L. Luskin, 2nd edn. John Wiley, New York.
  • Borodin, A. N. and Salminen, P. (2002). Handbook of Brownian Motion – Facts and Formulae, 2nd edn. Birkhäuser, Basel.
  • Carlier, G. (2003). Duality and existence for a class of mass transportation problems and economic applications. Adv. Math. Econom. 5, 1–22.
  • Cox, A. M. G., Hobson, D. and Obloj, J. (2011). Utility theory front to back – inferring utility from agents' choices. Preprint arXiv:1101.3572.
  • Cox, J. C. and Leland, H. E. (2000). On dynamic investment strategies. J. Econom. Dynamics and Control 24, 1859–1880.
  • Dayanik, S. and Karatzas, I. (2003). On the optimal stopping problem for one-dimensional diffusions. Stoch. Process. Appl. 107, 173–212.
  • Ekström, E. and Hobson, D. (2011). Recovering a time-homogeneous stock price process from perpetual option prices. Ann. Appl. Prob. 21, 1102–1135.
  • Gangbo, W. and McCann, R. J. (1996). The geometry of optimal transportation. Acta Math. 177, 113–161.
  • Gittins, J. C. and Glazebrook, K. D. (1977). On Bayesian models in stochastic scheduling. J. Appl. Prob. 14, 556–565.
  • 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.
  • He, H. and Huang, C. F. (1994). Consumption-portfolio policies: An inverse optimal problem. J. Econom. Theory 62, 257–293.
  • Hobson, D. and Klimmek, M. (2011). Constructing time-homogeneous generalized diffusions consistent with optimal stopping values. Stochastics 83, 477–503.
  • Jewitt, I. (1987). Risk aversion and the choice between risky prospects: The preservation of comparative statics results. Rev. Econom. Stud. 54, 73–85.
  • Karatzas, I. (1984). Gittins indices in the dynamic allocation problem for diffusion processes. Ann. Prob. 12, 173–192.
  • Klimmek, R. (1986). Risikoneigung und Besteuerung, University for business administration, Florentz, Munich.
  • Lu, B. (2010). Recovering a piecewise constant volatility from perpetual put option prices. J. Appl. Prob. 47, 680–692.
  • Milgrom, P. and Segal, I. (2002). Envelope theorems for arbitrary choice sets. Econometrica 70, 583–601.
  • Rogers, L. C. G. and Williams, D. (2000). Diffusions, Markov Processes and Martingales, Vol. 2. Cambridge University Press.
  • Rüschendorf, L. (1991). Fréchet-bounds and their applications. In Advances in Probability Distributions with Given Marginals: beyond the copulas, ed. G. Dall'Aglio. Kluwer Academic Publishers, Dordrecht.
  • Salminen, P. (1985). Optimal stopping of one-dimensional diffusions. Math. Nachr. 124, 85–101.
  • Samuelson, P. (1948). Consumption theory in terms of revealed preference. Economica 15, 243–253.
  • Samuelson, P. (1964). Tax deductibility of economic depreciation to insure invariant valuations. J. Political Econom. 72, 604–606.
  • Whittle, P. (1988). Restless bandits: Activity allocation in a changing world. J. Appl. Prob. 25A, 287–298.