Advances in Applied Probability

Bayesian quickest detection problems for some diffusion processes

Pavel V. Gapeev and Albert N. Shiryaev

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We study the Bayesian problems of detecting a change in the drift rate of an observable diffusion process with linear and exponential penalty costs for a detection delay. The optimal times of alarms are found as the first times at which the weighted likelihood ratios hit stochastic boundaries depending on the current observations. The proof is based on the reduction of the initial problems into appropriate three-dimensional optimal stopping problems and the analysis of the associated parabolic-type free-boundary problems. We provide closed-form estimates for the value functions and the boundaries, under certain nontrivial relations between the coefficients of the observable diffusion.

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Adv. in Appl. Probab., Volume 45, Number 1 (2013), 164-185.

First available in Project Euclid: 15 March 2013

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Zentralblatt MATH identifier

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60] 62M20: Prediction [See also 60G25]; filtering [See also 60G35, 93E10, 93E11] 34K10: Boundary value problems
Secondary: 62C10: Bayesian problems; characterization of Bayes procedures 62L15: Optimal stopping [See also 60G40, 91A60] 60J60: Diffusion processes [See also 58J65]

disorder detection diffusion process multidimensional optimal stopping stochastic boundary a change-of-variable formula with local time on surfaces parabolic-type free-boundary problem


Gapeev, Pavel V.; Shiryaev, Albert N. Bayesian quickest detection problems for some diffusion processes. Adv. in Appl. Probab. 45 (2013), no. 1, 164--185. doi:10.1239/aap/1363354107.

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  • Bayraktar, E. and Dayanik, S. (2006). Poisson disorder problem with exponential penalty for delay. Math. Operat. Res. 31, 217–233.
  • Bayraktar, E., Dayanik, S. and Karatzas, I. (2005). The standard Poisson disorder problem revisited. Stoch. Process. Appl. 115, 1437–1450.
  • Bayraktar, E., Dayanik, S. and Karatzas, I. (2006). Adaptive Poisson disorder problem. Ann. Appl. Prob. 16, 1190–1261.
  • Beibel, M. (2000). A note on sequential detection with exponential penalty for the delay. Ann. Statist. 28, 1696–1701.
  • Bensoussan, A. and Lions, J.-L. (1982). Applications of Variational Inequalities in Stochastic Control. North-Holland, Amsterdam.
  • Dayanik, S. (2010). Compound Poisson disorder problems with nonlinear detection delay penalty cost functions. Sequent. Anal. 29, 193–216.
  • Dayanik, S. and Sezer, S. O. (2006). Compound Poisson disorder problem. Math. Operat. Res. 31, 649–672.
  • Dynkin, E. B. (1963). The optimum choice of the instant for stopping a Markov process. Soviet Math. Dokl. 4, 627–629.
  • Friedman, A. (1976). Stochastic Differential Equations and Applications, Vol. 2. Academic Press, New York.
  • Gapeev, P. V. and Peskir, G. (2006). The Wiener disorder problem with finite horizon. Stoch. Process. Appl. 116, 1770–1791.
  • Gapeev, P. V. and Shiryaev, A. N. (2011). On the sequential testing problem for some diffusion processes. Stochastics 83, 519–535.
  • Grigelionis, B. I. and Shiryaev, A. N. (1966). On Stefan's problem and optimal stopping rules for Markov processes. Theory Prob. Appl. 11, 541–558.
  • Kolmogorov, A. N. (1992). On analytical methods in probability theory. In Selected Works of A. N. Kolmogorov, vol. II, ed. A. N. Shiryaev, Kluwer, Dordrecht, pp. 62–108.
  • Krylov, N. V. (1980). Controlled Diffusion Processes. Springer, New York.
  • Liptser, R. S. and Shiryaev, A. N. (1977). Statistics of Random Processes I. Springer, Berlin.
  • Øksendal, B. (1998). Stochastic Differential Equations. Springer, Berlin.
  • Peskir, G. (2007). A change-of-variable formula with local time on surfaces. In Séminaire de Probababilité XL (Lecture Notes Math. 1899). Springer, Berlin, pp. 69–96.
  • Peskir, G. and Shiryaev, A. N. (2006). Optimal Stopping and Free-Boundary Problems. Birkhäuser, Basel.
  • Poor, H. V. (1998). Quickest detection with exponential penalty for delay. Ann. Statist. 26, 2179–2205.
  • Poor, H. V. and Hadjiliadis, O. (2008). Quickest Detection. Cambridge University Press.
  • Pospisil, L., Vecer, J. and Hadjiliadis, O. (2009). Formulas for stopped diffusion processes with stopping times based on drawdowns and drawups. Stochastic Process. Appl. 119, 2563–2578.
  • Revuz, D. and Yor, M. (1999). Continuous Martingales and Brownian Motion. Springer, Berlin.
  • Shiryaev, A. N. (1961). The problem of the most rapid detection of a disturbance in a stationary process. Soviet Math. Dokl. 2, 795–799.
  • Shiryaev, A. N. (1963). On optimum methods in quickest detection problems. Theory Prob. Appl. 8, 22–46.
  • Shiryaev, A. N. (1964). On Markov sufficient statistics in nonadditive Bayes problems of sequential analysis. Theory Prob. Appl. 9, 670–686.
  • Shiryaev, A. N. (1965). Some exact formulas in a `disorder' problem. Theory Prob. Appl. 10, 348–354.
  • Shiryaev, A. N. (1967). Two problems of sequential analysis. Cybernetics 3, 63–69.
  • Shiryaev, A. N. (1978). Optimal Stopping Rules. Springer, Berlin.
  • Shiryaev, A. N. (2002). Quickest detection problems in the technical analysis of the financial data. In Mathematical Finance–-Bachelier Congress 2000 (Paris, June/July 2000), eds. H. Geman et al., Springer, Berlin, pp. 487–521.
  • Shiryaev, A. N. (2008). Generalized Bayesian nonlinear quickest detection problems: on Markov family of sufficient statistics. In Mathematical Control Theory and Finance (Lisbon, April 2007), eds. A. Sarychev et al., Springer, Berlin, pp. 377–386.
  • Shiryaev, A. N. and Zryumov, P. Y. (2009). On the linear and nonlinear generalized Bayesian disorder problem (discrete time case). In Optimality and Risk–-Modern Trends in Mathematical Finance, eds. F. Delbaen et al., Springer, Berlin, pp. 227–235.
  • Veretennikov, A. Yu. (1980). On the strong solutions of stochastic differential equations. Theory Prob. Appl. 24, 354–366.