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Teor. Veroyatnost. i Primenen., 2000, Volume 45, Issue 1, Pages 125–136 (Mi tvp327)  

This article is cited in 41 scientific papers (total in 41 papers)

Stopping Brownian motion without anticipation as close as possible to its ultimate maximum

S. E. Graversena, G. Peskira, A. N. Shiryaevb

a Institute of Mathematics, University of Aarhus, Denmark
b Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: Let $B=(B_t)_{0 \le t \le 1}$ be the standard Brownian motion started at 0, and let $S_t=\max_{ 0 \le r \le t} B_r$ for $0 \le t \le 1$. Consider the optimal stopping problem
$$ V_*=\inf_\tau{\mathsf E}(B_\tau-S_1)^2, $$
where the infimum is taken over all stopping times of $B$ satisfying $0 \le \tau \le 1$. We show that the infimum is attained at the stopping time $\tau_*=\inf\{0\le t\le 1\mid S_t-B_t\ge z_*\sqrt{1-t}\}$, where $z_*=1.12 \ldots$ is a unique root of the equation $4\Phi(z_*)-2z_*\varphi(z_*)-3=0$ with $\varphi(x)=(1/\sqrt{2 \pi }) e^{-x^2/2}$ and $ \Phi (x)=\int_{-\infty}^x \varphi(y) dy$. The value $V_*$ equals $2 \Phi (z_*)-1$. The method of proof relies upon a stochastic integral representation of $S_1$, time-change arguments, and the solution of a free-boundary (Stefan) problem.

Keywords: Brownian motion, optimal stopping, anticipation, ultimate maximum, free-boundary (Stefan) problem, Ito–Clark representation theorem, Markov process, diffusion.

DOI: https://doi.org/10.4213/tvp327

Full text: PDF file (471 kB)

English version:
Theory of Probability and its Applications, 2001, 45:1, 41–50

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Received: 21.10.1999

Citation: S. E. Graversen, G. Peskir, A. N. Shiryaev, “Stopping Brownian motion without anticipation as close as possible to its ultimate maximum”, Teor. Veroyatnost. i Primenen., 45:1 (2000), 125–136; Theory Probab. Appl., 45:1 (2001), 41–50

Citation in format AMSBIB
\by S.~E.~Graversen, G.~Peskir, A.~N.~Shiryaev
\paper Stopping Brownian motion without anticipation as close as possible to its ultimate maximum
\jour Teor. Veroyatnost. i Primenen.
\yr 2000
\vol 45
\issue 1
\pages 125--136
\jour Theory Probab. Appl.
\yr 2001
\vol 45
\issue 1
\pages 41--50

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    1. M. A. Urusov, “Optimal forecasting of the time of attaining the maximum by Brownian motion”, Russian Math. Surveys, 57:1 (2002), 163–164  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    2. Shiryaev A.N., “Quickest detection problems in the technical analysis of the financial data”, Mathematical Finance, Bachelier Congress, Springer Finance, 2002, 487–521  crossref  mathscinet  zmath  isi
    3. A. N. Shiryaev, M. Yor, “On the problem of stochastic integral representations of functionals of the Brownian motion. I”, Theory Probab. Appl., 48:2 (2004), 304–313  mathnet  crossref  crossref  mathscinet  zmath  isi
    4. M. A. Urusov, “On a property of the moment at which Brownian motion attains its maximum and some optimal stopping problems”, Theory Probab. Appl., 49:1 (2005), 169–176  mathnet  crossref  crossref  mathscinet  zmath  isi
    5. Renaud J.-F., Remillard B., “Explicit martingale representations for Brownian functionals and applications to option hedging”, Stochastic Analysis and Applications, 25:4 (2007), 801–820  crossref  mathscinet  zmath  isi  scopus
    6. Du Toit J., Peskir G., “The trap of complacency in predicting the maximum”, Annals of Probability, 35:1 (2007), 340–365  crossref  mathscinet  zmath  isi  scopus
    7. 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
    8. Shiryaev A., Xu Z., Zhou X.Yu., “Thou shalt buy and hold”, Quantitative Finance, 8:8 (2008), 765–776  crossref  mathscinet  zmath  isi  elib  scopus
    9. 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
    10. Rueschendorf L., Urusov M.A., “On a class of optimal stopping problems for diffusions with discontinuous coefficients”, Annals of Applied Probability, 18:3 (2008), 847–878  crossref  mathscinet  zmath  isi  scopus
    11. du Toit J., Peskir G., Shiryaev A.N., “Predicting the last zero of Brownian motion with drift”, Stochastics-An International Journal of Probability and Stochastic Processes, 80:2–3 (2008), 229–245  crossref  mathscinet  zmath  isi  scopus
    12. du Toit J., Peskir G., “Predicting the Time of the Ultimate Maximum for Brownian Motion with Drift”, Mathematical Control Theory and Finance, 2008, 95–112  crossref  mathscinet  zmath  isi  scopus
    13. Theory Probab. Appl., 54:1 (2010), 14–28  mathnet  crossref  crossref  mathscinet  isi
    14. 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
    15. Allaart P., “A General 'Bang-Bang' Principle for Predicting the Maximum of a Random Walk”, J Appl Probab, 47:4 (2010), 1072–1083  crossref  mathscinet  zmath  isi  elib  scopus
    16. Cohen A., “Examples of optimal prediction in the infinite horizon case”, Statistics & Probability Letters, 80:11–12 (2010), 950–957  crossref  mathscinet  zmath  isi  scopus
    17. 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
    18. Bernyk V., Dalang R.C., Peskir G., “Predicting the Ultimate Supremum of a Stable Levy Process with No Negative Jumps”, Ann Probab, 39:6 (2011), 2385–2423  crossref  mathscinet  zmath  isi  elib  scopus
    19. Dai M., Zhong Y., “Optimal Stock Selling/Buying Strategy with Reference to the Ultimate Average”, Mathematical Finance, 22:1 (2012), 165–184  crossref  mathscinet  zmath  isi  elib  scopus
    20. Theory Probab. Appl., 57:2 (2013), 357–366  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    21. Ano K., Ivanov R.V., “On Predicting the Ultimate Maximum for Exponential Levy Processes”, Electron. Commun. Probab., 17 (2012), 1–9  crossref  mathscinet  isi  scopus
    22. Allaart P.C., “Predicting the Supremum: Optimality of ‘Stop at Once Or Not at All’”, J. Appl. Probab., 49:3 (2012), 806–820  crossref  mathscinet  zmath  isi  elib  scopus
    23. Yam S.C.P., Yung S.P., Zhou W., “Optimal Selling Time in Stock Market Over a Finite Time Horizon”, Acta Math. Appl. Sin.-Engl. Ser., 28:3 (2012), 557–570  crossref  mathscinet  zmath  isi  elib  scopus
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    25. Dai M., Yang Zh., Zhong Y., “Optimal Stock Selling Based on the Global Maximum”, SIAM J. Control Optim., 50:4 (2012), 1804–1822  crossref  mathscinet  zmath  isi  elib  scopus
    26. 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
    27. Ekstrom E., Lindberg C., “Optimal Closing of a Momentum Trade”, J. Appl. Probab., 50:2 (2013), 374–387  crossref  mathscinet  zmath  isi  elib  scopus
    28. 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
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    30. 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
    31. 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
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    33. 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
    34. Fotopoulos S.B., Jandhyala V.K., Luo Yu., “Subordinated Brownian Motion: Last Time the Process Reaches Its Supremum”, Sankhya Ser. A, 77:1 (2015), 46–64  crossref  mathscinet  zmath  isi  scopus
    35. Perez I., Le H., “Time-Randomized Stopping Problems For a Family of Utility Functions”, SIAM J. Control Optim., 53:3 (2015), 1328–1345  crossref  mathscinet  zmath  isi  elib  scopus
    36. A. A. Kamenov, “Optimal stopping for absolute maximum of homogeneous diffusion”, Moscow University Mathematics Bulletin, 70:5 (2015), 202–207  mathnet  crossref  isi
    37. Baurdoux E.J., Kyprianou A.E., Ott C., “Optimal prediction for positive self-similar Markov?processes”, Electron. J. Probab., 21 (2016), 48  crossref  mathscinet  zmath  isi  elib  scopus
    38. Rokhlin D.B., “Minimax Perfect Stopping Rules For Selling An Asset Near Its Ultimate Maximum”, Optim. Lett., 11:8 (2017), 1743–1756  crossref  mathscinet  zmath  isi  scopus
    39. Liu Yu., Privault N., “A Recursive Algorithm For Selling At the Ultimate Maximum in Regime-Switching Models”, Methodol. Comput. Appl. Probab., 20:1 (2018), 369–384  crossref  mathscinet  zmath  isi  scopus
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    41. Guerra M. Nunes C. Oliveira C., “Optimal Stopping of One-Dimensional Diffusions With Integral Criteria”, J. Math. Anal. Appl., 481:2 (2020), 123473  crossref  mathscinet  isi
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