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Teor. Veroyatnost. i Primenen., 2004, Volume 49, Issue 1, Pages 184–190 (Mi tvp245)  

This article is cited in 17 scientific papers (total in 17 papers)

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On a property of the moment at which Brownian motion attains its maximum and some optimal stopping problems

M. A. Urusov

M. V. Lomonosov Moscow State University

Abstract: Let $B=(B_t)_{0\le t\le 1}$ be a standard Brownian motion and $\theta$ be the moment at which $B$ attains its maximal value, i.e., $B_\theta=\max_{0\le t\le 1}B_t$. Denote by $(\mathscr{F}^B_t)_{0\le t\le 1}$ the filtration generated by $B$. We prove that for any $(\mathscr{F}^B_t)$-stopping time $\tau$ $(0\le\tau\le 1)$, the following equality holds:
$$ E(B_\theta-B_\tau)^2=E|\theta-\tau|+\frac{1}{2}. $$
Together with the results of [S. E. Graversen, G. Peskir, and A. N. Shiryaev, Theory Probab. Appl., 45 (2000), pp. 41–50] this implies that the optimal stopping time $\tau_*$ in the problem
$$ \inf_\tauE|\theta-\tau| $$
has the form
$$ \tau_*=\inf\{0\le t\le 1: S_t-B_t\ge z_*\sqrt{1-t} \}, $$
where $S_t=\max_{0\le s\le t}B_s$, $z_*$ is a unique positive root of the equation $4\Phi(z)-2z\phi(z)-3=0$, $\phi(z)$ and $\Phi(z)$ are the density and the distribution function of a standard Gaussian random variable. Similarly, we solve the optimal stopping problems
$$ \inf_{\tau\in\mathfrak{M}_\alpha}E(\tau-\theta)^+ \quadand\quad \inf_{\tau\in\mathfrak{N}_\alpha}E(\tau-\theta)^-, $$
where $\mathfrak{M}_\alpha=\{\tau\colon E(\tau-\theta)^-\le \alpha\}$, and $\mathfrak{N}_\alpha=\{\tau\colon E(\tau-\theta)^+\le\alpha\}$. The corresponding optimal stopping times are of the same form as above (with other $z_*$'s).

Keywords: moment of attaining the maximum, Brownian motion, optimal stopping.

DOI: https://doi.org/10.4213/tvp245

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English version:
Theory of Probability and its Applications, 2005, 49:1, 169–176

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Received: 11.12.2003

Citation: M. A. Urusov, “On a property of the moment at which Brownian motion attains its maximum and some optimal stopping problems”, Teor. Veroyatnost. i Primenen., 49:1 (2004), 184–190; Theory Probab. Appl., 49:1 (2005), 169–176

Citation in format AMSBIB
\by M.~A.~Urusov
\paper On a property of the moment at which Brownian motion attains its maximum
and some optimal stopping problems
\jour Teor. Veroyatnost. i Primenen.
\yr 2004
\vol 49
\issue 1
\pages 184--190
\jour Theory Probab. Appl.
\yr 2005
\vol 49
\issue 1
\pages 169--176

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    This publication is cited in the following articles:
    1. A. N. Shiryaev, “On Conditional-Extremal Problems of the Quickest Detection of Nonpredictable Times of the Observable Brownian Motion”, Theory Probab. Appl., 53:4 (2009), 663–678  mathnet  crossref  crossref  mathscinet  zmath  isi
    2. Fotopoulos S.B., Hu X., Munson C.L., “Flexible supply contracts under price uncertainty”, European Journal of Operational Research, 191:1 (2008), 253–263  crossref  mathscinet  zmath  isi  scopus
    3. du Toit J., Peskir G., Shiryaev A.N., “Predicting the last zero of Brownian motion with drift”, Stochastics, 80:2-3 (2008), 229–245  crossref  mathscinet  zmath  isi  elib  scopus
    4. du Toit J., Peskir G., “Predicting the time of the ultimate maximum for Brownian motion with drift”, Mathematical control theory and finance, Springer, Berlin, 2008, 95–112  crossref  mathscinet  zmath  isi  scopus
    5. du Toit J., Peskir G., “Selling a stock at the ultimate maximum”, Ann. Appl. Probab., 19:3 (2009), 983–1014  crossref  mathscinet  zmath  isi  scopus
    6. S. S. Sinelnikov, “On optimal stopping for Brownian motion with a negative drift”, Theory Probab. Appl., 56:2 (2011), 343–350  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    7. Bernyk V., Dalang R.C., Peskir G., “Predicting the ultimate supremum of a stable Lévy process with no negative jumps”, Ann. Probab., 39:6 (2011), 2385–2423  crossref  mathscinet  zmath  isi  elib  scopus
    8. A. K. Rozov, A. N. Tsarapkin, “Metod obratnoi induktsii v zadachakh obnaruzheniya spontanno voznikayuschikh yavlenii”, Vestn. S.-Peterburg. un-ta. Ser. 10. Prikl. matem. Inform. Prots. upr., 2012, no. 3, 88–97  mathnet
    9. Peskir G., “Optimal Detection of a Hidden Target: the Median Rule”, Stoch. Process. Their Appl., 122:5 (2012), 2249–2263  crossref  mathscinet  zmath  isi  elib  scopus
    10. Espinosa G.-E. Touzi N., “Detecting the Maximum of a Scalar Diffusion with Negative Drift”, SIAM J. Control Optim., 50:5 (2012), 2543–2572  crossref  mathscinet  zmath  isi  elib  scopus
    11. Glover K. Hulley H. Peskir G., “Three-Dimensional Brownian Motion and the Golden Ratio Rule”, Ann. Appl. Probab., 23:3 (2013), 895–922  crossref  mathscinet  zmath  isi  elib  scopus
    12. Glover K. Hulley H., “Optimal Prediction of the Last-Passage Time of a Transient Diffusion”, SIAM J. Control Optim., 52:6 (2014), 3833–3853  crossref  mathscinet  isi  scopus
    13. Baurdoux E.J. van Schaik K., “Predicting the Time At Which a Levy Process Attains Its Ultimate Supremum”, Acta Appl. Math., 134:1 (2014), 21–44  crossref  mathscinet  zmath  isi  scopus
    14. Peskir G., “Quickest Detection of a Hidden Target and Extremal Surfaces”, Ann. Appl. Probab., 24:6 (2014), 2340–2370  crossref  mathscinet  zmath  isi  scopus
    15. Elie R. Espinosa G.-E., “Optimal Selling Rules For Monetary Invariant Criteria: Tracking the Maximum of a Portfolio With Negative Drift”, Math. Financ., 25:4 (2015), 754–788  crossref  mathscinet  zmath  isi  elib  scopus
    16. Ren D., “Optimal Stopping For the Last Exit Time”, Bull. Aust. Math. Soc., 99:1 (2019), 148–160  crossref  mathscinet  zmath  isi  scopus
    17. Ankirchner S. Kazi-Tani N. Klein M. Kruse T., “Stopping With Expectation Constraints: 3 Points Suffice”, Electron. J. Probab., 24 (2019), 66  crossref  isi
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