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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 HS5379.2006.1 and RGSF 060291821 a/G, Basic Research
Program of RAS Presidium 15, NSF grants DMI0300121 and DMI0600538 and NYSTAR, RFBR
grants 080100740 and 080191205YaF, and School of Mathematics (Manchester University).
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 Fall '11
 EugeneA.Feinberg
 Applied Mathematics

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