The Annals of Applied Probability

Quickest detection problems for Bessel processes

Peter Johnson and Goran Peskir

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Abstract

Consider the motion of a Brownian particle that initially takes place in a two-dimensional plane and then after some random/unobservable time continues in the three-dimensional space. Given that only the distance of the particle to the origin is being observed, the problem is to detect the time at which the particle departs from the plane as accurately as possible. We solve this problem in the most uncertain scenario when the random/unobservable time is (i) exponentially distributed and (ii) independent from the initial motion of the particle in the plane. The solution is expressed in terms of a stopping time that minimises the probability of a false early detection and the expected delay of a missed late detection.

Article information

Source
Ann. Appl. Probab., Volume 27, Number 2 (2017), 1003-1056.

Dates
Received: September 2015
Revised: March 2016
First available in Project Euclid: 26 May 2017

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1495764373

Digital Object Identifier
doi:10.1214/16-AAP1223

Mathematical Reviews number (MathSciNet)
MR3655860

Zentralblatt MATH identifier
1370.60135

Subjects
Primary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60] 60J60: Diffusion processes [See also 58J65] 60H30: Applications of stochastic analysis (to PDE, etc.)
Secondary: 35K67: Singular parabolic equations 45G10: Other nonlinear integral equations 62C10: Bayesian problems; characterization of Bayes procedures

Keywords
Quickest detection Brownian motion Bessel process optimal stopping parabolic partial differential equation free-boundary problem smooth fit entrance boundary nonlinear Fredholm integral equation the change-of-variable formula with local time on curves/surfaces

Citation

Johnson, Peter; Peskir, Goran. Quickest detection problems for Bessel processes. Ann. Appl. Probab. 27 (2017), no. 2, 1003--1056. doi:10.1214/16-AAP1223. https://projecteuclid.org/euclid.aoap/1495764373


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References

  • [1] Assing, S., Jacka, S. and Ocejo, A. (2014). Monotonicity of the value function for a two-dimensional optimal stopping problem. Ann. Appl. Probab. 24 1554–1584.
  • [2] Bayraktar, E., Dayanik, S. and Karatzas, I. (2005). The standard Poisson disorder problem revisited. Stochastic Process. Appl. 115 1437–1450.
  • [3] De Angelis, T. and Peskir, G. (2015). Global $C^{1}$ regularity of the value function in optimal stopping problems. Research Report No. 6, Probab. Statist. Group Manchester. To appear.
  • [4] du Toit, J. and Peskir, G. (2009). Selling a stock at the ultimate maximum. Ann. Appl. Probab. 19 983–1014.
  • [5] Engelbert, H.-J. and Peskir, G. (2014). Stochastic differential equations for sticky Brownian motion. Stochastics 86 993–1021.
  • [6] Feller, W. (1952). The parabolic differential equations and the associated semi-groups of transformations. Ann. of Math. (2) 55 468–519.
  • [7] Ferreyra, G. and Sundar, P. (2000). Comparison of solutions of stochastic equations and applications. Stoch. Anal. Appl. 18 211–229.
  • [8] Gapeev, P. V. and Peskir, G. (2006). The Wiener disorder problem with finite horizon. Stochastic Process. Appl. 116 1770–1791.
  • [9] Gapeev, P. V. and Shiryaev, A. N. (2013). Bayesian quickest detection problems for some diffusion processes. Adv. in Appl. Probab. 45 164–185.
  • [10] Josephy, M. (1981). Composing functions of bounded variation. Proc. Amer. Math. Soc. 83 354–356.
  • [11] Lieberman, G. M. (1996). Second Order Parabolic Differential Equations. World Scientific, River Edge, NJ.
  • [12] Peskir, G. (2005). On the American option problem. Math. Finance 15 169–181.
  • [13] 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.
  • [14] Peskir, G. (2015). Continuity of the optimal stopping boundary for two-dimensional diffusions. Probab. Statist. Group Manchester. Research Report No. 4. To appear.
  • [15] Peskir, G. and Shiryaev, A. (2006). Optimal Stopping and Free-Boundary Problems. Birkhäuser, Basel.
  • [16] Revuz, D. and Yor, M. (1999). Continuous Martingales and Brownian Motion, 3rd ed. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 293. Springer, Berlin.
  • [17] Rogers, L. C. G. and Williams, D. (2000). Diffusions, Markov Processes, and Martingales: Itô Calculus. Vol. 2. Cambridge Univ. Press, Cambridge.
  • [18] Shiryaev, A. N. (1961). The problem of the most rapid detection of a disturbance in a stationary process. Sov. Math., Dokl. 2 795–799.
  • [19] Shiryaev, A. N. (2010). Quickest detection problems: Fifty years later. Sequential Anal. 29 345–385.
  • [20] Shiryayev, A. N. (1978). Optimal Stopping Rules. Springer, New York.
  • [21] Volkonskiĭ, V. A. (1958). Random substitution of time in strong Markov processes. Theory Probab. Appl. 3 310–326.