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Phase-Type Distributions and Optimal Stopping for Autoregressive Processes

Part of: Sequential methods Stochastic processes

Published online by Cambridge University Press:  04 February 2016

Sören Christensen
Sören Christensen*
Affiliation:
Christian-Albrechts-Universität zu Kiel
*
∗ Postal address: Mathematisches Seminar, Christian-Albrechts-Universität zu Kiel, Ludewig-Meyn-Str. 4, 24098 Kiel, Germany. Email address: christensen@math.uni-kiel.de
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Abstract

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Autoregressive processes are intensively studied in statistics and other fields of applied stochastics. For many applications, the overshoot and the threshold time are of special interest. When the upward innovations are in the class of phase-type distributions, we determine the joint distribution of these two quantities and apply this result to problems of optimal stopping. Using a principle of continuous fit, this leads to explicit solutions.

Keywords

Autoregressive processthreshold timephase-type innovationoptimal stopping

MSC classification

Primary: 60G99: None of the above, but in this section
Secondary: 60G40: Stopping times; optimal stopping problems; gambling theory 62L15: Optimal stopping
Type
Research Article
Information
Journal of Applied Probability , Volume 49 , Issue 1 , March 2012 , pp. 22 - 39
Copyright
© Applied Probability Trust 

References

Asmussen, S. (2003). Applied Probability and Queues (Appl. Math. 51), 2nd edn. Springer, New York.Google Scholar
Christensen, S. and Irle, A. (2009). A note on pasting conditions for the American perpetual optimal stopping problem. Statist. Prob. Lett. 79, 349–353.CrossRefGoogle Scholar
Christensen, S., Irle, A. and Novikov, A. (2011). An elementary approach to optimal stopping problems for AR(1) sequences. Sequent. Anal. 30, 79–93.CrossRefGoogle Scholar
Feller, W. (1966). An Introduction to Probability Theory and Its Applications, Vol. II. John Wiley, New York.Google Scholar
Finster, M. (1982). Optimal stopping on autoregressive schemes. Ann. Prob. 10, 745–753.CrossRefGoogle Scholar
Frisén, M. and Sonesson, C. (2006). Optimal surveillance based on exponentially weighted moving averages. Sequent. Anal. 25, 379–403.CrossRefGoogle Scholar
Gasper, G. and Rahman, M. (2004). Basic Hypergeometric Series (Encyclopedia Math. Appl. 96), 2nd edn. Cambridge University Press.CrossRefGoogle Scholar
Goldie, C. M. and Maller, R. A. (2000). Stability of perpetuities. Ann. Prob. 28, 1195–1218.CrossRefGoogle Scholar
Jacobsen, M. and Jensen, A. T. (2007). Exit times for a class of piecewise exponential Markov processes with two-sided Jumps. Stoch. Process. Appl. 117, 1330–1356.CrossRefGoogle Scholar
Mikhalevich, V. S. (1958). Bayesian choice between two hypotheses for the mean value of a normal process. Visnik Kiiv. Univ. 1, 101–104.Google Scholar
Novikov, A. A. (2009). {On distributions of first passage times and optimal stopping of AR(1) sequences}. Theory Prob. Appl. 53, 419–429.CrossRefGoogle Scholar
Novikov, A. and Kordzakhia, N. (2008). Martingales and first passage times of AR(1) sequences. Stochastics 80, 197–210.CrossRefGoogle Scholar
Novikov, A. A. and Shiryaev, A. N. (2004). On an effective case of the solution of the optimal stopping problem for random walks. Teor. Veroyat. Primen. 49, 373–382 (in Russian). English translation: Theory Prob. Appl. 49 (2005), 344-354.Google Scholar
Novikov, A. and Shiryaev, A. (2007). On solution of the optimal stopping problem for processes with independent increments. Stochastics 79, 393–406.CrossRefGoogle Scholar
Peškir, G. and Shiryaev, A. N. (2000). Sequential testing problems for Poisson processes. Ann. Statist. 28, 837–859.CrossRefGoogle Scholar
Peškir, G. and Shiryaev, A. (2006). Optimal Stopping and Free-Boundary Problems. Birkhäuser, Basel.Google Scholar
Runger, G. C. and Prabhu, S. S. (1996). A Markov chain model for the multivariate exponentially weighted moving averages control chart. J. Amer. Statist. Assoc. 91, 1701–1706.CrossRefGoogle Scholar