Electronic Journal of Probability

Parameter-Dependent Optimal Stopping Problems for One-Dimensional Diffusions

Peter Bank and Christoph Baumgarten

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Abstract

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1464819849

Digital Object Identifier
doi:10.1214/EJP.v15-835

Mathematical Reviews number (MathSciNet)
MR2745722

Zentralblatt MATH identifier
1226.60057

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

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

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

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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References

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