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Disorder problem for a Brownian motion on a segment in the case of uniformly distributed moment of disorder

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Abstract

The paper deals with the quickest detection of the disorder of a Brownian motion on a finite interval. The unknown moment of disorder is assumed to be uniformly distributed on the interval. The Bayesian and absolute criteria are used as optimality tests. The problem is reduced to a classic optimal stopping problem where the optimal stopping time may be obtained as a solution to integral equations. The existence and uniqueness of the solution of integral equations are proved analytically.

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References

  1. E. Carlstein, H. G. Müller, andD. Siegmund, Change-Point Problems. IMS Lect. Notes Monogr. Ser. 23 (Springer, 1994).

  2. A. N. Shiryaev, “Quickest Detection Problems in the Technical Analysis of the Financial Data,” in Math. Finance Bachelier Congress (Springer, P., 2000), pp. 487-521.

  3. A. N. Shiryaev, Stochastic Sequential Analysis. Optimal Stopping Rules, 2nd ed. (Fizmatlit, Nauka, Moscow, 1976).

  4. G. Peskir and A. N. Shiryaev, Optimal Stopping and Free-Boundary Problems (Birkhäuser, Basel, 2006).

  5. P. V. Gapeev and G. Peskir, “The Wiener Disorder Problem with Finite Horizon,” Stochast. Process. Appl. 116 (12), 1770 (2006).

    Article  MathSciNet  Google Scholar 

  6. A. N. Shiryaev, “A Remark on the Quickest Detection Problems,” Statistics and Decisions 22, 79 (2004).

    Article  MathSciNet  Google Scholar 

  7. R. Sh. Liptser and A. N. Shiryaev, Statistics of Random, Processes (Nauka, Moscow, 1974) [in Russian].

  8. A. N. Shiryaev, Essentials of Stochastic Financial Mathematics, Vol. 1, 2 (Fasis, Moscow, 1998, 2004) [in Russian].

  9. A. N. Shiryaev, Probabilistic and Statistical Methods in Decision Theory (MCCME, FMEE, Moscow, 2011) [in Russian].

  10. G. Peskir, “A-Change-of-Variable Formula with Local Time on Curves,” J. Theor. Probab. 18 (3), 499 (2005).

    Article  MathSciNet  Google Scholar 

  11. M. Yor, “On Some Exponential Functionals of Brownian Motion,” Adv. Appl. Probab. 24, 509 (1992).

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Faculty of Mechanics and Mathematics, Moscow State University, Leninskie Gory, Moscow, 119991, Russia

    A. A. Sokko

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  1. A. A. Sokko
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Correspondence to A. A. Sokko.

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Original Russian Text © A.A. Sokko, 2015, published in Vestnik Moskovskogo Universiteta, Matematika. Mekhanika, 2015, Vol. 70, No. 3, pp. 3–11.

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Sokko, A.A. Disorder problem for a Brownian motion on a segment in the case of uniformly distributed moment of disorder. Moscow Univ. Math. Bull. 70, 103–110 (2015). https://doi.org/10.3103/S0027132215030018

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  • DOI: https://doi.org/10.3103/S0027132215030018

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