The Annals of Applied Probability

Quickest detection of a hidden target and extremal surfaces

Goran Peskir

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Abstract

Let $Z=(Z_{t})_{t\ge0}$ be a regular diffusion process started at $0$, let $\ell$ be an independent random variable with a strictly increasing and continuous distribution function $F$, and let $\tau_{\ell}=\inf\{t\ge0\vert Z_{t}=\ell\}$ be the first entry time of $Z$ at the level $\ell$. We show that the quickest detection problem

\[\inf_{\tau}[\mathsf{P}(\tau<\tau_{\ell})+c\mathsf{E}(\tau -\tau_{\ell})^{+}]\]

is equivalent to the (three-dimensional) optimal stopping problem

\[\sup_{\tau}\mathsf{E}[R_{\tau}-\int_{0}^{\tau}c(R_{t})\,dt],\]

where $R=S-I$ is the range process of $X=2F(Z)-1$ (i.e., the difference between the running maximum and the running minimum of $X$ ) and $c(r)=cr$ with $c>0$. Solving the latter problem we find that the following stopping time is optimal:

\[\tau_{*}=\inf \{t\ge0\vert f_{*}(I_{t},S_{t})\le X_{t}\le g_{*}(I_{t},S_{t})\},\]

where the surfaces $f_{*}$ and $g_{*}$ can be characterised as extremal solutions to a couple of first-order nonlinear PDEs expressed in terms of the infinitesimal characteristics of $X$ and $c$. This is done by extending the arguments associated with the maximality principle [Ann. Probab. 26 (1998) 1614–1640] to the three-dimensional setting of the present problem and disclosing the general structure of the solution that is valid in all particular cases. The key arguments developed in the proof should be applicable in similar multi-dimensional settings.

Article information

Source
Ann. Appl. Probab., Volume 24, Number 6 (2014), 2340-2370.

Dates
First available in Project Euclid: 26 August 2014

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1409058034

Digital Object Identifier
doi:10.1214/13-AAP979

Mathematical Reviews number (MathSciNet)
MR3262505

Zentralblatt MATH identifier
1338.60115

Subjects
Primary: 60G35: Signal detection and filtering [See also 62M20, 93E10, 93E11, 94Axx] 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60] 60J60: Diffusion processes [See also 58J65]
Secondary: 34A34: Nonlinear equations and systems, general 35R35: Free boundary problems 49J40: Variational methods including variational inequalities [See also 47J20]

Keywords
Quickest detection hidden target optimal stopping diffusion process maximum process minimum process range process excursion the maximality principle extremal surface the principle of smooth fit nonlinear differential equation

Citation

Peskir, Goran. Quickest detection of a hidden target and extremal surfaces. Ann. Appl. Probab. 24 (2014), no. 6, 2340--2370. doi:10.1214/13-AAP979. https://projecteuclid.org/euclid.aoap/1409058034


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References

  • [1] Barles, G., Daher, C. and Romano, M. (1994). Optimal control on the $L^{\infty}$ norm of a diffusion process. SIAM J. Control Optim. 32 612–634.
  • [2] Barron, E. N. and Ishii, H. (1989). The Bellman equation for minimizing the maximum cost. Nonlinear Anal. 13 1067–1090.
  • [3] Bernyk, V., Dalang, R. C. and Peskir, G. (2011). Predicting the ultimate supremum of a stable Lévy process with no negative jumps. Ann. Probab. 39 2385–2423.
  • [4] Cohen, A. (2010). Examples of optimal prediction in the infinite horizon case. Statist. Probab. Lett. 80 950–957.
  • [5] Cox, A. M. G., Hobson, D. and Obłój, J. (2008). Pathwise inequalities for local time: Applications to Skorokhod embeddings and optimal stopping. Ann. Appl. Probab. 18 1870–1896.
  • [6] du Toit, J. and Peskir, G. (2007). The trap of complacency in predicting the maximum. Ann. Probab. 35 340–365.
  • [7] du Toit, J. and Peskir, G. (2008). Predicting the time of the ultimate maximum for Brownian motion with drift. In Mathematical Control Theory and Finance 95–112. Springer, Berlin.
  • [8] du Toit, J. and Peskir, G. (2009). Selling a stock at the ultimate maximum. Ann. Appl. Probab. 19 983–1014.
  • [9] du Toit, J., Peskir, G. and Shiryaev, A. N. (2008). Predicting the last zero of Brownian motion with drift. Stochastics 80 229–245.
  • [10] Dubins, L. E., Gilat, D. and Meilijson, I. (2009). On the expected diameter of an $L_{2}$-bounded martingale. Ann. Probab. 37 393–402.
  • [11] Dubins, L. E. and Schwarz, G. (1988). A sharp inequality for sub-martingales and stopping-times. Astérisque 129–145.
  • [12] Dubins, L. E., Shepp, L. A. and Shiryaev, A. N. (1993). Optimal stopping rules and maximal inequalities for Bessel processes. Theory Probab. Appl. 38 226–261.
  • [13] Elie, R. and Espinosa, G.-E. (2014). Optimal selling rules for monetary invariant criteria: Tracking the maximum of a portfolio with negative drift. Math. Finance. To appear.
  • [14] Espinosa, G.-E. and Touzi, N. (2012). Detecting the maximum of a scalar diffusion with negative drift. SIAM J. Control Optim. 50 2543–2572.
  • [15] Gapeev, P. V. (2006). Discounted optimal stopping for maxima in diffusion models with finite horizon. Electron. J. Probab. 11 1031–1048 (electronic).
  • [16] Gapeev, P. V. (2007). Discounted optimal stopping for maxima of some jump-diffusion processes. J. Appl. Probab. 44 713–731.
  • [17] Glover, K., Hulley, H. and Peskir, G. (2013). Three-dimensional Brownian motion and the golden ratio rule. Ann. Appl. Probab. 23 895–922.
  • [18] Goldman, M. B., Sosin, H. B. and Gatto, M. A. (1979). Path dependent options: “Buy at the low, sell at the high.” J. Finance 34 1111–1127.
  • [19] Graversen, S. E., Peskir, G. and Shiryaev, A. N. (2001). Stopping Brownian motion without anticipation as close as possible to its ultimate maximum. Theory Probab. Appl. 45 41–50.
  • [20] Guo, X. and Zervos, M. (2010). $\pi$ options. Stochastic Process. Appl. 120 1033–1059.
  • [21] Heinricher, A. C. and Stockbridge, R. H. (1991). Optimal control of the running max. SIAM J. Control Optim. 29 936–953.
  • [22] Hobson, D. (2007). Optimal stopping of the maximum process: A converse to the results of Peskir. Stochastics 79 85–102.
  • [23] Jacka, S. D. (1991). Optimal stopping and best constants for Doob-like inequalities. I. The case $p=1$. Ann. Probab. 19 1798–1821.
  • [24] Obłój, J. (2004). The Skorokhod embedding problem and its offspring. Probab. Surv. 1 321–390.
  • [25] Obłój, J. (2007). The maximality principle revisited: On certain optimal stopping problems. In Séminaire de Probabilités XL. Lecture Notes in Math. 1899 309–328. Springer, Berlin.
  • [26] Pedersen, J. L. (2000). Discounted optimal stopping problems for the maximum process. J. Appl. Probab. 37 972–983.
  • [27] Pedersen, J. L. (2003). Optimal prediction of the ultimate maximum of Brownian motion. Stoch. Stoch. Rep. 75 205–219.
  • [28] Peskir, G. (1998). Optimal stopping of the maximum process: The maximality principle. Ann. Probab. 26 1614–1640.
  • [29] Peskir, G. (2005). Maximum process problems in optimal control theory. J. Appl. Math. Stoch. Anal. 1 77–88.
  • [30] Peskir, G. (2007). A change-of-variable formula with local time on surfaces. In Séminaire de Probabilités XL. Lecture Notes in Math. 1899 69–96. Springer, Berlin.
  • [31] Peskir, G. (2012). A duality principle for the Legendre transform. J. Convex Anal. 19 609–630.
  • [32] Peskir, G. (2012). Optimal detection of a hidden target: The median rule. Stochastic Process. Appl. 122 2249–2263.
  • [33] Peskir, G. and Shiryaev, A. (2006). Optimal Stopping and Free-Boundary Problems. Lectures in Mathematics ETH Zürich. Birkhäuser, Basel.
  • [34] Shiryaev, A. N. (1961). The problem of the most rapid detection of a disturbance of a stationary regime. Sov. Math. Dokl. 2 795–799.
  • [35] Shiryaev, A. N. (1963). On optimal methods in quickest detection problems. Theory Probab. Appl. 8 22–46.
  • [36] Shiryaev, A. N. (2002). Quickest detection problems in the technical analysis of the financial data. In Mathematical Finance—Bachelier Congress, 2000 (Paris). Springer Finance 487–521. Springer, Berlin.
  • [37] Shiryaev, A. N. (2009). On conditional-extremal problems of the quickest detection of nonpredictable times of the observable Brownian motion. Theory Probab. Appl. 53 663–678.
  • [38] Shiryaev, A. N. and Novikov, A. A. (2008). On a stochastic version of the trading rule “buy and hold”. Statist. Decisions 26 289–302.
  • [39] Urusov, M. A. (2004). On a property of the moment at which Brownian motion attains its maximum and some optimal stopping problems. Theory Probab. Appl. 49 169–176.
  • [40] Zhitlukhin, M. (2009). A maximal inequality for skew Brownian motion. Statist. Decisions 27 261–280.