TVP2009 - c 2009 Society for Industrial and Applied...

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Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. T HEORY P ROBAB. A PPL . c ± 2009 Society for Industrial and Applied Mathematics Vol. 53, No. 3, pp. 519–536 Translated from Russian Journal ON ASYMPTOTIC OPTIMALITY OF THE SECOND ORDER IN THE MINIMAX QUICKEST DETECTION PROBLEM OF DRIFT CHANGE FOR BROWNIAN MOTION E. V. BURNAEV ,E .A .FE INBERG , AND A. N. SHIRYAEV § ( Translated by E. V. Burnaev ) Abstract. This paper deals with the minimax quickest detection problem of a drift change for the Brownian motion. The following minimax risks are studied: C ( T )=inf τ M T sup θ E θ ( τ θ | τ = θ )and C ( T )=in f τ M T sup θ E θ ( τ θ | τ = θ ), where M T is the set of stopping times τ such that E τ = T and M T is the set of randomized stopping times τ such that E τ = T . The goal of this paper is to obtain for these risks estimates from above and from below. Using these estimates we prove the existence of stopping times, which are asymptotically optimal of the ±rst and second orders as T →∞ (for C ( T C ( T ), respectively). Key words. disorder problem, Brownian motion, minimax risk, asymptotical optimality of the ±rst and second orders DOI. 10.1137/S0040585X97983791 1. Introduction. Problem formulation and discussion of results. 1.1. The Bayesian formulation of the quickest detection problem of drift change for Brownian motion (“disorder problem”) was given in [6], [7]. In these papers the solution of this problem was given for the case of the exponential a priori distribution of the disorder time and using a special passage to the limit , a solution of the disorder problem in the generalized Bayesian setting was obtained (the time when the “dis- order” appears is interpreted as a generalized random variable with the “uniform” distribution on R + =[0 , )). Let us remark that the problem of the quickest detec- tion of a disturbance of a stationary regime is reduced to the generalized Bayesian disorder problem (see [7], [21, subsection 4.2], [22, section 3.9]). The direct solution of the generalized Bayesian problem without passage to the limit was obtained in [11] (Variant (B)). In [11, Theorem 3.1] estimates from above and from below for minimax risk were also obtained. The minimax risk characterizes the detection delay of the disorder time (Variant (C)). In order to decrease the “gap” between the upper and lower bounds for the min- imax risk (see inequalities (13) below and inequalities (3.1) in [11]), in the present paper we consider (for giving an alarm signal that the disorder has appeared) not only Markov times (stopping times), which are deFned by the history of the observed process, but randomized stopping times as well (Variant ( C)). The main result of this work is the following. We prove that in the class of the randomized stopping times there exists the stopping time, which is asymptotically Received by the editors November 8, 2007. This work was supported by Analytical Departmen- tal Program RNP.2.2.1.1.2467, grants HS-5379.2006.1 and RGSF 06-02-91821 a/G, Basic Research Program of RAS Presidium 15, NSF grants DMI-0300121 and DMI-0600538 and NYSTAR, RFBR grants 08-01-00740 and 08-01-91205-YaF, and School of Mathematics (Manchester University).
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TVP2009 - c 2009 Society for Industrial and Applied...

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