Sequential testing problems for Bessel processes
HTML articles powered by AMS MathViewer
- by Peter Johnson and Goran Peskir PDF
- Trans. Amer. Math. Soc. 370 (2018), 2085-2113
Abstract:
Consider the motion of a Brownian particle that takes place either in a two-dimensional plane or in three-dimensional space. Given that only the distance of the particle to the origin is being observed, the problem is to detect the true dimension as soon as possible and with minimal probabilities of the wrong terminal decisions. We solve this problem in the Bayesian formulation under any prior probability of the true dimension when the passage of time is penalised linearly.References
- Jacques du Toit and Goran Peskir, Selling a stock at the ultimate maximum, Ann. Appl. Probab. 19 (2009), no. 3, 983–1014. MR 2537196, DOI 10.1214/08-AAP566
- P. V. Gapeev and G. Peskir, The Wiener sequential testing problem with finite horizon, Stoch. Stoch. Rep. 76 (2004), no. 1, 59–75. MR 2038029, DOI 10.1080/10451120410001663753
- Pavel V. Gapeev and Albert N. Shiryaev, On the sequential testing problem for some diffusion processes, Stochastics 83 (2011), no. 4-6, 519–535. MR 2842593, DOI 10.1080/17442508.2010.530349
- Peter Johnson and Goran Peskir, Quickest detection problems for Bessel processes, Ann. Appl. Probab. 27 (2017), no. 2, 1003–1056. MR 3655860, DOI 10.1214/16-AAP1223
- V. S. Mikhalevich, A Bayes test of two hypotheses concerning the mean of a normal process, Visn. Kiiv. Univ. 1 (1958), 254–264.
- Goran Peskir, On the American option problem, Math. Finance 15 (2005), no. 1, 169–181. MR 2116800, DOI 10.1111/j.0960-1627.2005.00214.x
- Goran Peskir, A change-of-variable formula with local time on curves, J. Theoret. Probab. 18 (2005), no. 3, 499–535. MR 2167640, DOI 10.1007/s10959-005-3517-6
- Goran Peskir and Albert Shiryaev, Optimal stopping and free-boundary problems, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2006. MR 2256030
- Daniel Revuz and Marc Yor, Continuous martingales and Brownian motion, 3rd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 293, Springer-Verlag, Berlin, 1999. MR 1725357, DOI 10.1007/978-3-662-06400-9
- L. C. G. Rogers and David Williams, Diffusions, Markov processes, and martingales. Vol. 2, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2000. Itô calculus; Reprint of the second (1994) edition. MR 1780932, DOI 10.1017/CBO9781107590120
- A. N. Shiryaev, Two problems of sequential analysis, Cybernetics 3 (1967), no. 2, 63–69 (1969). MR 272119, DOI 10.1007/BF01078755
- A. N. Shiryayev, Optimal stopping rules, Applications of Mathematics, Vol. 8, Springer-Verlag, New York-Heidelberg, 1978. Translated from the Russian by A. B. Aries. MR 0468067
- V. A. Volkonskiĭ, Random substitution of time in strong Markov processes, Teor. Veroyatnost. i Primenen 3 (1958), 332–350 (Russian, with English summary). MR 0100919
- Abraham Wald, Sequential Analysis, John Wiley & Sons, Inc., New York; Chapman & Hall, Ltd., London, 1947. MR 0020764
Additional Information
- Peter Johnson
- Affiliation: School of Mathematics, The University of Manchester, Oxford Road, Manchester M13 9PL, United Kingdom
- Email: peter.johnson-3@manchester.ac.uk
- Goran Peskir
- Affiliation: School of Mathematics, The University of Manchester, Oxford Road, Manchester M13 9PL, United Kingdom
- MR Author ID: 337521
- Email: goran@maths.man.ac.uk
- Received by editor(s): June 7, 2016
- Received by editor(s) in revised form: September 8, 2016
- Published electronically: September 8, 2017
- Additional Notes: This research was supported by a grant from the British Engineering and Physical Sciences Research Council (EPSRC)
- © Copyright 2017 by the authors
- Journal: Trans. Amer. Math. Soc. 370 (2018), 2085-2113
- MSC (2010): Primary 60G40, 60J60, 60H30; Secondary 35K10, 45G10, 62C10
- DOI: https://doi.org/10.1090/tran/7068
- MathSciNet review: 3739203