Geodesic
On Conditional-Extremal Problems of the Quickest Detection of Nonpredictable Times of the Observable Brownian Motion
Teoriâ veroâtnostej i ee primeneniâ, Tome 53 (2008) no. 4, pp. 751-768 Cet article a éte moissonné depuis la source Math-Net.Ru

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We consider nonpredictable stopping times $\theta=\inf\{t\le 1:B_t=\max_{0\le s\le 1}B_s\}$, $g=\sup\{t\le 1:B_t=0\}$ for the Brownian motion $B=(B_t)_{0\le t\le 1}$. The main results of the paper concern solving the following conditional-extremal problems: in classes of Markov times $\mathfrak{M}_\alpha^B(\theta)=\{\tau\le 1:P\,\{\tau<\theta\}\le\alpha\}$, $\mathfrak{M}_\alpha^B(g)=\{\sigma\le 1:P\,\{\sigma, where $0<\alpha<1$, to describe a structure of optimal stopping times $\tau_\alpha^*(\theta)$ and $\sigma_\alpha^*(g)$, for which $E\,[\tau_\alpha^*(\theta)-\theta]^+=\inf_{\tau\in\mathfrak{M}_\alpha^B(\theta)}E\,(\tau-\theta)^+$, $E\,[\sigma_\alpha^*(g)-g]^+=\inf_{\sigma\in\mathfrak{M}_\alpha^B(g)}E\,(\sigma-g)^+$. We also consider the problems of the structure of some special stopping times in the classes $\mathfrak{M}_\alpha^B(\theta^\mu)$ and $\mathfrak{M}_\alpha^B(g^\mu)$ for the case of Brownian motion with drift $B^\mu=(B_t^\mu)_{0\le t\le 1}$, where $B_t^\mu=\mu t+B_t$.
Keywords: conditional-extremal problems, nonpredictable time, quickest detection, Brownian motion. 
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     title = {On {Conditional-Extremal} {Problems} of the {Quickest} {Detection} of {Nonpredictable} {Times} of the {Observable} {Brownian} {Motion}},
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A. N. Shiryaev. On Conditional-Extremal Problems of the Quickest Detection of Nonpredictable Times of the Observable Brownian Motion. Teoriâ veroâtnostej i ee primeneniâ, Tome 53 (2008) no. 4, pp. 751-768. http://geodesic.mathdoc.fr/item/TVP_2008_53_4_a5/

[1] Graversen S. E., Peskir G., Shiryaev A. N., “Stopping Brownian motion without anticipation as close as possible to its ultimate maximum”, Teoriya veroyatn. i ee primen., 45:1 (2000), 125–136 | MR | Zbl

[2] Pedersen J. L., “Optimal prediction of the ultimate maximum of Brownian motion”, Stochastics Stochastics Rep., 75:4 (2003), 205–219 | DOI | MR | Zbl

[3] Shiryaev A. N., “Quickest detection problems in the technical analysis of the financial data”, Mathematical Finance, Bachelier congress 2000, Springer Finance, eds. H. Geman et al., Springer-Verlag, Berlin, 2002, 487–521 | MR | Zbl

[4] Urusov M. A., “Ob odnom svoistve momenta dostizheniya maksimuma brounovskim dvizheniem i nekotorykh zadachakh optimalnoi ostanovki”, Teoriya veroyatn. i ee primen., 49:1 (2004), 184–190 | MR | Zbl

[5] Peskir G., Shiryaev A. N., Optimal Stopping and Free-Boundary Problems, Birkhäuser, Basel, 2006, 500 pp. | MR | Zbl

[6] Shiryaev A. N., Statisticheskii posledovatelnyi analiz, Nauka, M., 1969 | MR

[7] Zhakod Zh., Shiryaev A. N., Predelnye teoremy dlya sluchainykh protsessov, v. 1, 2, Nauka, M., 1994, 544 pp.; Теория вероятн. и матем. статист., 47, 48, 368 с.

[8] du Toit J., Peskir G., “Predicting the time of the ultimate maximum for Brownian motion with drift”, Proceedings of the Workshop on Mathematical Control Theory and Finance (Lisbon, 2007), Springer-Verlag, Berlin, 2008, 95–112 | MR | Zbl

[9] Shiryaev A. N., Osnovy stokhasticheskoi finansovoi matematiki, T. 1: Fakty, modeli; T. 2: Teoriya, FAZIS, M., 1998, 1018 pp.

[10] Revuz D., Yor M., Continuous Martingales and Brownian Motion, Grundlehren Math. Wiss., 293, Springer-Verlag, Berlin, 1999, 602 pp. | MR | Zbl