Electronic Journal of Probability

Parameter-Dependent Optimal Stopping Problems for One-Dimensional Diffusions

Peter Bank and Christoph Baumgarten

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We consider a class of optimal stopping problems for a regular one-dimensional diffusion whose payoff depends on a linear parameter. As shown in Bank and Föllmer (2003) problems of this type may allow for a universal stopping signal that characterizes optimal stopping times for any given parameter via a level-crossing principle of some auxiliary process. For regular one-dimensional diffusions, we provide an explicit construction of this signal in terms of the Laplace transform of level passage times. Explicit solutions are available under certain concavity conditions on the reward function. In general, the construction of the signal at a given point boils down to finding the infimum of an auxiliary function of one real variable. Moreover, we show that monotonicity of the stopping signal corresponds to monotone and concave (in a suitably generalized sense) reward functions. As an application, we show how to extend the construction of Gittins indices of Karatzas (1984) from monotone reward functions to arbitrary functions.

Article information

Electron. J. Probab., Volume 15 (2010), paper no. 64, 1971-1993.

Accepted: 26 November 2010
First available in Project Euclid: 1 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60]
Secondary: 60J60: Diffusion processes [See also 58J65] 91G20: Derivative securities

Optimal stopping Gittins index multi-armed bandit problems American options universal stopping signal

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Bank, Peter; Baumgarten, Christoph. Parameter-Dependent Optimal Stopping Problems for One-Dimensional Diffusions. Electron. J. Probab. 15 (2010), paper no. 64, 1971--1993. doi:10.1214/EJP.v15-835. https://projecteuclid.org/euclid.ejp/1464819849

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  • Bank, P. and Föllmer, H. American options, multi-armed bandits, and optimal consumption plans: a unifying view. Paris-Princeton Lectures on Mathematical Finance, 2002, 1–42, Lecture Notes in Math., 1814, Springer, Berlin, 2003.
  • Bank, P. and El Karoui, N. A stochastic representation theorem with applications to optimization and obstacle problems. Ann. Probab. 32 (2004), no. 1B, 1030–1067.
  • Beibel, M. ; Lerche, H. R. A note on optimal stopping of regular diffusions under random discounting. Teor. Veroyatnost. i Primenen. 45 (2000), no. 4, 657–669; translation in Theory Probab. Appl. 45 (2002), no. 4, 547–557.
  • Dayanik, S. and Karatzas, I. On the optimal stopping problem for one-dimensional diffusions. Stochastic Process. Appl. 107 (2003), no. 2, 173–212.
  • Dayanik, S. Optimal stopping of linear diffusions with random discounting. Math. Oper. Res. 33 (2008), no. 3, 645–661.
  • Dynkin, E. B. Optimal choice of the stopping moment of a Markov process. (Russian) Dokl. Akad. Nauk SSSR 150 1963 238–240.
  • Dynkin, E. B. Markov processes. Vols. I, II. Translated with the authorization and assistance of the author by J. Fabius, V. Greenberg, A. Maitra, G. Majone. Die Grundlehren der Mathematischen Wissenschaften, Bände 121, 122 Academic Press Inc., Publishers, New York; Springer-Verlag, Berlin-Göttingen-Heidelberg 1965 Vol. I: xii+365 pp.; Vol. II: viii+274 pp.
  • El Karoui, N. Les aspects probabilistes du contrôle stochastique. (French) [The probabilistic aspects of stochastic control] Ninth Saint Flour Probability Summer School 1979 (Saint Flour, 1979), pp. 73–238, Lecture Notes in Math., 876, Springer, Berlin-New York, 1981.
  • El Karoui, N. and Föllmer, H. A non-linear Riesz representation in probabilistic potential theory. Ann. Inst. H. Poincaré Probab. Statist. 41 (2005), no. 3, 269–283.
  • Fakeev, A. G. On the question of the optimal stopping of a Markov process. (Russian) Teor. Verojatnost. i Primenen. 16 (1971), 708–710.
  • Gittins, J. C. Bandit processes and dynamic allocation indices. With discussion. J. Roy. Statist. Soc., Ser. B 41 (1979), no. 2, 148–177.
  • Gittins, J. C. and Glazebrook, K. D. On Bayesian models in stochastic scheduling. J. Appl. Probability 14 (1977), no. 3, 556–565.
  • Gittins, J. C. and Jones, D. A dynamic allocation index for the discounted multiarmed bandit problem. Biometrika 66 (1977), no. 3, 556–565.
  • Itô, K.; McKean, H.P., Jr. Diffusion processes and their sample paths. Second printing, corrected. Die Grundlehren der mathematischen Wissenschaften, Band 125. Springer-Verlag, Berlin-New York, 1974. xv+321 pp.
  • Johnson, T.C. ; Zervos, M. The solution to a second order linear ordinary differential equation with a non-homogeneous term that is a measure. Stochastics 79 (2007), no. 3-4, 363–382.
  • Karatzas, I. Gittins indices in the dynamic allocation problem for diffusion processes. Ann. Probab. 12 (1984), no. 1, 173–192.
  • Karatzas, I. On the pricing of American options. Appl. Math. Optim. 17 (1988), no. 1, 37–60.
  • Karatzas, I. and Shreve, S.E. Brownian motion and stochastic calculus. Second edition. Graduate Texts in Mathematics, 113. Springer-Verlag, New York, 1991. xxiv+470 pp. ISBN: 0-387-97655-8
  • Kaspi, H. and Mandelbaum, A. Multi-armed bandits in discrete and continuous time. Ann. Appl. Probab. 8 (1998), no. 4, 1270–1290.
  • Peskir, G. and Shiryaev, A. Optimal stopping and free-boundary problems. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 2006. xxii+500 pp. ISBN: 978-3-7643-2419-3; 3-7643-2419-8
  • Revuz, D. and Yor, M. Continuous martingales and Brownian motion. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 293. Springer-Verlag, Berlin, 1991. x+533 pp. ISBN: 3-540-52167-4
  • Shiryayev, A. N. Optimal stopping rules. Translated from the Russian by A. B. Aries. Applications of Mathematics, Vol. 8. Springer-Verlag, New York-Heidelberg, 1978. x+217 pp. ISBN: 0-387-90256-2