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Unformatted text preview: Quickest detection of drift change for Brownian
motion in generalized Bayesian and minimax
settings
Eugene A. Feinberg, Albert N. Shiryaev
Received: November 17, 2006; Accepted: February 14, 2007 Summary: The paper deals with the quickest detection of a change of the drift of the Brownian
motion. We show that the generalized Bayesian formulation of the quickest detection problem
can be reduced to the optimal stopping problem for a diffusion Markov process. For this problem
the optimal procedure is described and its characteristics are found. We show also that the same
procedure is asymptotically optimal for the minimax formulation of the quickest detection problem. 1 Introduction and problem formulation
1. This paper deals with the problem of the quickest detection of a change of a drift for
a Brownian motion formulated and studied in Shiryaev [17, 18, 19] in the Bayesian and
generalized Bayesian settings. Let B = ( Bt )t ≥0 be a standard Brownian motion deﬁned
on a ﬁltered probability space ( , F , (Ft )t ≥0 , P). Without loss of generality we shall
assume that is the space of continuous functions ω = (ωt )t ≥0. Everywhere in this
paper, the equality between two random variable deﬁned on this probability space means
that these random variables are equal P a.s.
Suppose we observe a stochastic process X = ( X t )t ≥0 that has the following structure:
X t = µ(t − θ)+ + σ Bt (1.1) or, equivalently,
dX t = σ dBt ,
µ dt + σ dBt , t < θ,
t ≥ θ, with X 0 = 0. Here µ and σ are known numbers, where µ = 0 and σ > 0, and θ is an
unknown time; θ ∈ [0, ∞]. We interpret θ < ∞ as a time when the “disorder” appears,
and θ = ∞ means that the disorder never happens. The appearance of a disorder should
be detected as soon as possible trying to avoid false alarms.
AMS 2000 subject classiﬁcation: Primary: 60G35, 60G40; Secondary: 93E10
Key words and phrases: Brownian motion, disorder, generalized Bayesian and minimax formulations of the
quickest detection problem, optimal stopping, asymptotical optimality This article is protected by German copyright law. You may copy and distribute this article for your personal use only. Other use is only allowed with written permission by the copyright holder. Statistics & Decisions 24, 445–470 (2006) / DOI 10.1524/stnd.2006.24.4.445
c Oldenbourg Wissenschaftsverlag, M¨ nchen 2006
u Let Pθ = Law( X θ) be the distribution of the process X under the assumption that the
disorder happened at the deterministic time θ. In particular, P∞ is the distribution of X
under the assumption that the disorder never happened, i.e. P∞ = Law(σ Bt , t ≥ 0), and
P0 is the distribution of the process µt + σ Bt , t ≥ 0.
Let τ = τ(ω) be a ﬁnite stopping stopping time with respect to the ﬁltration F X =
X)
(Ft t ≥0 generated by the process X t . We interpret τ as the decision that the disorder has
happened at the time τ(ω).
If the system is controlled by a stopping time τ , occurrences of false alarms can
be characterized in several ways. For example, they can be characterized either by the
probability of the false alarm
αθ = Pθ (τ < θ), θ ∈ [0, ∞), or by the mean time until the false alarm
T = E ∞ τ,
or by their combinations, where E θ denotes the expectation with respect to the probability
Pθ , θ ∈ [0, ∞].
2. Recall the following two variants (A) and (B) of the problem of quickest detection,
the socalled Bayesian and generalized Bayesian formulations, proposed in [17, 18, 19].
Variant (A). Suppose that θ is a random variable, θ = θ(ω), independent of B =
( Bt )t ≥0 and having an exponential distribution with an atom at 0,
P(θ = 0) = π,
P(θ > t  θ > 0) = e−λt ,
where π ∈ [0, 1) and λ > 0 are known constants. For a constant α ∈ (0, 1) denote
M(α) = {τ : P(τ < θ) ≤ α}. Variant (A) of the quickest detection problem is to ﬁnd, for
∗
a given α ∈ (0, 1), a stopping time τ(α) , if it exists, such that
∗
∗
E (τ(α) − θ  τ(α) ≥ θ) = inf E (τ − θ  τ ≥ θ).
τ ∈M(α) Variant (B). According to this variant, θ is a parameter in [0, ∞] rather than a random
variable considered in Variant (A). For every T > 0 we denote by
M T = {τ : E ∞ τ = T }
the set of stopping times with the P∞ mean T. Variant (B) is, for a given T ∈ (0, ∞), to
∗
ﬁnd a stopping time τT , if it exists, such that
∞
0 ∗
E θ (τT − θ)+ dθ = inf τ ∈M T ∞
0 E θ (τ − θ)+ dθ. (1.2) This variant of the quickest detection problem is called generalized Bayesian, since θ
can be interpreted as a generalized random variable with the “uniform” distribution on
[0, ∞). This article is protected by German copyright law. You may copy and distribute this article for your personal use only. Other use is only allowed with written permission by the copyright holder. Feinberg  Shiryaev 446 447 Remark 1.1 It is natural to consider a bigger class MT = {τ : T ≤ E ∞ τ < ∞} ⊃ MT
and to try to ﬁnd a policy τT such that
¯∗
∞
0 E θ (τT − θ)+ dθ = inf
¯∗ τ ∈M T ∞
0 E θ (τ − θ)+ dθ. (1.3) However, we shall see in Corollary 4.3 below that the inﬁma on the righthand sides of
equations (1.2) and (1.3) are equal and therefore there is no advantage in considering the
class MT instead of the class MT .
As was shown in [17, 18, 19], there is an optimal detection stopping time for Variant (A) and it can be described in the following way. Let
πt = P θ ≤ t  FtX
be the a posteriori probability that the disorder has appeared before time t. In particular,
π0 = π. The stopping time
∗
τ(α) = inf {t ≥ 0 : πt ≥ 1 − α} (1.4) is optimal in the class M(α) .
The process (πt )t ≥0 satisﬁes the stochastic differential equation
dπt = λ − µ2 2
µ
π (1 − πt ) dt + 2 πt (1 − πt ) dX t
σ2 t
σ with π0 = π ; see [17, 18, 19] for details. Set
ϕt = πt
.
1 − πt According to Itˆ ’s formula
o
dϕt = λ(1 + ϕt ) dt + µ
ϕt dX t
σ2 (1.5) with ϕ0 = π0 /(1 − π0 ).
Let
Lt = d P0 FtX
d P∞ FtX be a Radon–Nikod´ m derivative, also called the likelihood, of P0 FtX with respect to
y
P∞ FtX . It is well known [9, 21] that
µ Lt = e σ2 Xt − 1
2 µ2
t
σ2 This article is protected by German copyright law. You may copy and distribute this article for your personal use only. Other use is only allowed with written permission by the copyright holder. Quickest detection and by Itˆ ’s formula
o
dL t = µ
L t dX t ,
σ2 L 0 = 1. (1.6) By using equation (1.6) and applying Itˆ ’s formula to the function λeλt L t
o
0
we ﬁnd the following representation for the strong solution ϕt of equation (1.5): t t ϕt = λ
0 eλ t L t
du ,
eλu L u du
eλu L u , (1.7) when ϕ0 = 0. In a general case,
t ϕ t = ϕ 0 eλ t L t + λ
0 eλt L t
du .
eλu L u Denote
ψt (λ) = ϕt
.
λ Then, if ϕ0 = 0, we see from (1.7) that
t ψt (λ) =
0 eλt L t
du
eλu L u (1.8) and from (1.5) we ﬁnd that
dψt (λ) = (1 + λψt (λ)) dt + µ
ψt (λ) dX t .
σ2 (1.9) ∗
Using the processes (ϕt )t ≥0 and (ψt (λ))t ≥0 , the stopping time τ(α) deﬁned by (1.4) can
be presented as
∗
τ(α) = inf t ≥ 0 : ϕt ≥ 1−α
α and, in the case π0 = ϕ0 = 0,
∗
τ(α) = inf t ≥ 0 : ψt (λ) ≥ 1−α
.
αλ (1.10) Following [17, 18, 19], let
λ → 0, α → 1, (1.11) 1−α
→ T > 0,
λ (1.12) and where T is a ﬁxed constant. This article is protected by German copyright law. You may copy and distribute this article for your personal use only. Other use is only allowed with written permission by the copyright holder. Feinberg  Shiryaev 448 449 Equations (1.8) and (1.9) imply that under (1.11) and (1.12) the limit
ψt = lim ψt (λ)
λ→0 exists, allows the representation
t ψt =
0 Lt
du ,
Lu (1.13) and satisﬁes the equation
dψt = dt + µ
ψt dX t ,
σ2 ψ0 = 0. (1.14) Moreover, under (1.11) and (1.12), we deﬁne the stopping time
∗
∗
τT = lim τ(α) .
α → 1,
λ→0 This stopping time has the following representation:
∗
τT = inf {t ≥ 0 : ψt ≥ T }. (1.15) It is interesting and important for our further considerations to notice that
∗
E ∞ τT = T. (1.16) Indeed, under the measure P∞ the differential equation (1.14) has the form
dψt = dt + µ
ψt dBt ,
σ ψ0 = 0. (1.17) ∗
Since ψt ∧τT ≤ T , by the martingale properties of stochastic integrals, E∞ ∗
τT ψt dBt = 0. 0 So, from (1.17)
∗
∗
T = E ∞ ψτT = E ∞ τT . In other words, T = lim 1−α , where the limit is taken as indicated in (1.11) and (1.12),
λ
has a simple meaning of the expected time until the process (ψt∗ )t ≥0 reaches the level T ,
taken under the assumption that a disorder never happens.
It was shown in [17, 18, 19] that, under the limits (1.11) and (1.12), the optimal
∗
stopping times τ(α) for the Bayesian Variant (A) converge to an optimal stopping time for
the generalized Bayesian problem formulated in Variant (B). Namely, the stopping time
∗
∗
τT = limα→1,λ→0 τ(α) is optimal in the sense of (1.2).
3. One of the main results of the present paper is the direct proof in Section 2 of
∗
the optimality of the stopping time τT = inf {t ≥ 0 : ψt ≥ T } in Variant (B). This This article is protected by German copyright law. You may copy and distribute this article for your personal use only. Other use is only allowed with written permission by the copyright holder. Quickest detection direct proof clariﬁes why the process (ψt )t ≥0 is a sufﬁcient statistics for this problem.
(For the discrete time case, the corresponding method is called the “Shiryaev–Roberts”
procedure.) This direct proof is also useful for our analysis in Section 3 of the quickest
detection problem in the following formulation.
∗
Variant (C). If it exists, in the class MT = {τ : E ∞ τ = T } ﬁnd a stopping time σT
for which
∗
∗
sup E θ (σT − θ  σT ≥ θ) = inf sup E θ (τ − θ  τ ≥ θ),
θ ≥0 (1.18) τ ∈M T θ ≥ 0 where, similarly to Variant (B), θ ∈ [0, ∞).
Although optimal policies for this minimax problem are still unknown, we shall see
in Section 3 that our approach, based on the results for Variant (B), implies that, at least
∗
for large T , the stopping time τT = inf {t ≥ 0 : ψt ≥ T } is asymptotically optimal.
We remark that asymptotic optimality has been established in the statistical literature
for many change point models; see, for example, collection of papers [2]. In addition to
asymptotic optimality, our approach leads to twosided inequalities (3.5) for the value
function on the righthand side of (1.18). These inequalities provide estimates how close
the performance of the asymptotically optimal stopping time is to the optimal value.
We shall discuss another interesting and popular minimax formulation of the quickest
detection problem, Variant (D), in Section 5. 2 Variant (B)
1. Since for any stopping time τ
(τ − θ)+ = ∞
θ I(u ≤ τ) du , we observe that
E θ (τ − θ)+ = ∞
θ E θ I(u ≤ τ) du = ∞
θ E∞ Lu
I(u ≤ τ) du = E ∞
Lθ τ
θ Lu
du ,
Lθ
(2.1) where the second equality in (2.1) holds since for θ ≤ u
X
d Pθ Fu
dPθ
I(u ≤ τ) = E u
I (u ≤ τ)
X
dPu
d Pu Fu
Lu
Lu
= E u I (u ≤ τ) = E ∞ I (u ≤ τ) .
Lθ
Lθ E θ I(u ≤ τ) = E u (2.2) Here the ﬁrst equality in (2.2) follows from the property Pθ
Pu , the second equality
X
follows from {u ≤ τ } ∈ Fu , and the last equality follows from the property Pu ( A) =
X
P∞ ( A) for all A ∈ Fu . To prove the third equality in (2.2), we notice that for θ ≤ u
X
d Pθ Fu
X
d Pu Fu = X
d Pθ Fu
X
d P∞ Fu = X
d Pθ Fu
X
d ( P0 Fu ) X
d ( P∞ Fu )
X
d ( P0 Fu ) = Lu
X
d ( P0 Fu )
X
d ( Pθ Fu ) = Lu
,
Lθ This article is protected by German copyright law. You may copy and distribute this article for your personal use only. Other use is only allowed with written permission by the copyright holder. Feinberg  Shiryaev 450 451 where the equality
X
d P0 Fu = Lθ X
d Pθ Fu (2.3) X
follows from the speciﬁc property of the model (1.1) that P0 Fu is the measure of
X is the measure for the process X =
the process X t = µt + σ Bt , t ≤ u , and Pθ Fu
t
0 µI(s ≥ θ) ds + σ Bt , t ≤ u . Since the increments of these two processes coincide
on the time interval [θ, u ], the Radon–Nikod´ m derivative of the measure for the ﬁrst
y
´
process with respect to the measure for the second process is equal to the Radon–Nikodym
derivative of the measure for the process X t = µt + σ Bt , t ≤ θ , with respect to the measure
for the process X t = σ Bt , t ≤ θ. Thus,
X
d P0 Fu = X
d Pθ Fu d P0 FθX
d Pθ FθX = Lθ . We remark that formula (2.3) also follows from direct calculations of the Radon–Nikod´ m
y
derivative in (2.3) by applying, for example, the Corollary from [9, Theorem 7.18].
After integrating in θ the ﬁrst and the last expressions in (2.1), we ﬁnd that
∞
0 ∞ E θ (τ − θ)+ dθ = E ∞ = E∞ τ
0 u
0 τ
θ 0 Lu
dθ du = E ∞
Lθ Lu
du dθ
Lθ
τ (2.4) ψu du , 0 where
u ψu =
0 Lu
dθ.
Lθ Hence, the following statement holds.
Lemma 2.1 (a) For any stopping time τ
∞
0 E θ (τ − θ)+ dθ = E ∞ τ ψu du . (2.5) 0 (b) The value
B(T ) = inf τ ∈M T 1
T ∞
0 E θ (τ − θ)+ dθ of the generalized Bayesian problem equals to the value of the conditionalextremal
optimal stopping problem for the process (ψt )t ≥0 :
B(T ) = inf τ ∈M T 1
E∞
T τ
0 ψu du . (2.6) This article is protected by German copyright law. You may copy and distribute this article for your personal use only. Other use is only allowed with written permission by the copyright holder. Quickest detection ∗
In Section 4 we shall show that the optimal stopping time for this problem is τT ,
deﬁned in (1.15), that prescribes to stop when the process (ψt )t ≥0 hits the level T. So, B( T ) =
2. To compute E ∞ ∗
τT
0 ∗
τT 1
E∞
T ψu du . (2.7) 0 ψu du , we introduce the function
∗
τT (x
U(x ) = E ∞) ψu du , (2.8) 0 (x) (x) where E ∞ is the expectation for the distribution P∞ for the Markov process (ψt )t ≥0
satisfying the stochastic differential equation (1.17) with ψ0 = x , x ∈ [0, ∞).
The inﬁnitesimal operator L ∞ of this Markov process is
L∞ = ∂
∂2
+ ρx 2 2 ,
∂x
∂x with the “signal to noise” ratio ρ =
X
d P0 Fu
X
d ( P∞ Fu ) µ2
.
2σ 2 x > 0, Note, by the way, that the Kullback–Leibler X
X
between the measures P0 Fu and P∞ Fu is exactly ρu .
divergence E 0 log
The function U = U(x ), x ∈ (0, T ), satisﬁes Kolmogorov’s backward equation L ∞ U(x ) = −x ,
i.e.
U + ρx 2 U = −x ; (2.9) see [3] for details on this equation.
To compute the function U = U(x ), x ∈ [0, T ], we introduce for x ≥ 0 the function
F(x ) = ex (− Ei(−x )), (2.10) where
∞ − Ei(−x ) =
x e−u
du ,
u is the integral exponential function [4, 5]. Let
∞ G (x ) =
x F(u )
du .
u2 (2.11) Solutions of the problems considered in this paper depend on µ and σ only via the
µ2
“signal to noise” ratio ρ = 2σ 2 . Without loss of generality we shall primarily consider
in this paper the case ρ = 1 and formulate statements of theorems and lemmas for this
case. In Section 4 we shall explain how the major characteristics should be recalculated
when ρ = 1. Under ρ = 1 we have the following result. This article is protected by German copyright law. You may copy and distribute this article for your personal use only. Other use is only allowed with written permission by the copyright holder. Feinberg  Shiryaev 452 453 Lemma 2.2 For all T > 0 and x ∈ [0, T ]
U(x ) = G 1
T 1
.
x −G (2.12) In particular, for T > 0 and x = 0
U(0) = G 1
T . (2.13) ∗
Proof: According to (1.16), E ∞ τT = T. Thus, for all 0 ≤ x ≤ T U(x ) = ∗
τT (x
E ∞) 0 (x ∗
(0 ∗
∗
ψs ds ≤ TE ∞)τT ≤ TE ∞)τT = TE ∞ τT = T 2 . Therefore, U(x ) is a bounded solution of equation (2.9) with the boundary condition
U(T ) = 0.
It is easy to ﬁnd all the bounded solutions of equation (2.9). These solutions are
U(x ) = C1 − ∞
1
x eu
u2 ∞
u e−z
dz
z du = C 1 − ∞
1
x F(u )
1
du = C1 − G
.
2
x
u 1
The boundary condition U(T ) = 0 implies C1 = G ( T ). This gives (2.12). Since
G (z ) → 0 as z → ∞, (2.12) implies (2.13). 3. Set b = 1
T.
∞ G (b) =
b By (2.13), U(0) = G (b). Integrating (2.11) by parts, we ﬁnd
∞ F(u )
du = −
u2 F(u ) d b F(b)
1
=
+
u
b ∞
b F(u )
1
du − ,
u
b (2.14) where we used the formula
F (u ) = F(u ) − 1
,
u that follows directly from (2.10).
Formula (2.14) implies that
bG (b) = F(b) − (b), (2.15) F(u )
du .
u (2.16) where
∞ (b) = 1 − b
b Since
∞ − Ei(−u ) =
u e−t
dt = e−u
t ∞
0 e−t
dt
u+t This article is protected by German copyright law. You may copy and distribute this article for your personal use only. Other use is only allowed with written permission by the copyright holder. Quickest detection and F(u ) = eu (− Ei(−u )), we ﬁnd that
∞
b ∞ F(u )
du =
u
∞ =
0 ∞ e−t
b ∞
∞
eu (− Ei(−u ))
e−t
du =
dt du
u
u (u + t)
b
b
0
t
∞
∞
log 1 + b
du
log(1 + u )
e−t
e−bu
dt =
dt =
du .
u (u + t)
t
u
0
0 Hence,
∞ (b) = 1 − b e−bu 0 log(1 + u )
du .
u (2.17) Taking into account that
B( T ) = 1
E∞
T ∗
τT ψu du = 0 1
1
1
U(0) = G
T
T
T = bG (b), we have that formulae (2.15)–(2.17) imply the following result.
Theorem 2.3 In the Variant (B), B(T ) = F(b) −
∞ B(T ) = F(b) − 1 − b
b (b), where b = 1/T , and thus
∞ F(u )
du = F(b) − 1 − b
u e−bu b log(1 + u )
du .
u
(2.18) 4. Let us study the asymptotics of B(T ) for the case ρ = 1 when T → ∞ and when
∗
T → 0. We recall that the policy τT deﬁned in (1.15) is optimal in the class MT .
Theorem 2.4 In the Variant (B),
B( T ) = log2 T
T log T − 1 − C + O
T
2 + O(T 2 ), , T → ∞, (2.19) T → 0, where C = 0.577 . . . is Euler’s constant. Proof: We shall use the following known formulas [8, (3.1.6) and (3.2.4)] for the integral
exponential function: for all b > 0
− Ei(−b) = log 1
−C+
b ∞
n =1 (−1)n+1 bn
n · n! (2.20) and for large b > 0 and any n ≥ 1
− Ei(−b) = e−b 1
+
b n (−1)k
k=1 k!
1
+ O n+1
bk+1
b . (2.21) This article is protected by German copyright law. You may copy and distribute this article for your personal use only. Other use is only allowed with written permission by the copyright holder. Feinberg  Shiryaev 454 455 First, we consider the case of small T , i.e. b is large. By (2.21), with n = 1, we have
1
1
+O 2
b
b F(b) = eb (− Ei(−b)) = = T + O(T 2 ). (2.22) Let us show now that for b → ∞
1
1
+O 2 .
2b
b (b) = (2.23) Formula (2.23) is important because (2.22) and (2.23) imply that for b → ∞
B(T ) = F(b) − (b) = 1
1
+O 2
2b
b = T
+ O(T 2 ).
2 (2.24) To prove (2.23), we write
∞
0 log(1 + u )
du
u
1
log(1 + u )
e−bu
=
du +
u
0
e−bu (2.25)
2 e−bu 1 log(1 + u )
du +
u ∞ e−bu 2 log(1 + u )
du .
u Since log(1 + u ) ≤ log(2u − 1) for u ≥ 2,
∞ e−bu 2 log(1 + u )
du ≤
u ∞ e−bu 1 log(2u − 1)
1
b
du =
− Ei −
u
2
2 2 , (2.26) where the last expression follows from the Laplace transform expression [4, p. 148] of
the function
0,
0 < u < 1,
log(2u − 1), u ≥ 1. f (u ) = (2.27) From (2.26) and from (2.21) with n = 0, for b → ∞
∞ b e−bu 2 log(1 + u )
1
b
du ≤ b ·
Ei −
u
2
2 2 = 2e−b
1
1+O 2
b
b =o 1
.
b2
(2.28) Next, for b → ∞
2 b e−bu 1 Also, since
1 b
0 1
u e−bu log(1 + u )
du ≤ be−b
u log(1 + u ) = 1 −
log(1 + u )
du ≤ b
u u
2 2 log(1 + u ) du < be−b = o 1 1
.
b2 (2.29) + O(u 2 ), for 0 < u ≤ 1 and for b → ∞
1
0 e−bu 1 − u
1
1
+ O(u 2 ) du = 1 −
+O 2 .
2
2b
b
(2.30) This article is protected by German copyright law. You may copy and distribute this article for your personal use only. Other use is only allowed with written permission by the copyright holder. Quickest detection Hence, for b → ∞, formulae (2.25)–(2.30) imply
∞ (b) = 1 − b e−bu 0 log(1 + u )
1
1
du =
+O 2 .
u
2b
b (2.31) Thus, (2.23) and therefore (2.24) are proved.
Finally, consider the case of large T , i.e. b is small. By (2.10) and (2.20), the ﬁrst term
in (2.18) has the following asymptotic as b → 0:
F(b) = eb (− Ei(−b)) = log 1
1
− C + O b log
b
b log T
T = log T − C + O .
(2.32) The remaining part of the last expression in (2.18) can be estimated in the following way.
Since log(1 + u ) ≤ u for u ≥ 0, for b → 0
2 b
0 e−bu log(1 + u )
du ≤ b
u 2 e−bu du ≤ 2b = O(b) = O 0 1
T . (2.33) From (2.26) and (2.20) we have for b → 0
∞ b
2 e−bu log(1 + u )
b
b
du ≤
Ei −
u
2
2 2 = O b log2 1
b =O log2 T
T . (2.34) Hence, from (2.31)–(2.34) we conclude that
B(T ) = F(b) − ∞ (b) = F(b) − 1 + b
0 = F(b) − 1 + O(b) + O b log2 1
b e−bu log(1 + u )
du
u = log T − 1 − C + O log2 T
T . 3 Variant (C)
1. The quickest detection problems in Variant (C) are popular in the statistical and quality
control literature, especially in the case of discrete time; see [2, 6, 22] and references
therein. We investigate a continuoustime version of Variant (C) for scheme (1.1) in
this section. The following theorem provides upper and lower estimates for C(T ) =
inf τ ∈MT supθ ≥0 E θ (τ − θ  τ ≥ θ).
∗
Theorem 3.1 For T > 0 and for τT = inf {t ≥ 0 : ψt ≥ T } B(T ) ≤ C(T ) ≤ C ∗ (T ),
∗
where C ∗ (T ) = E 0 τT . (3.1) This article is protected by German copyright law. You may copy and distribute this article for your personal use only. Other use is only allowed with written permission by the copyright holder. Feinberg  Shiryaev 456 457 Lemma 3.3 below implies that
∗
C ∗ ( T ) = E 0 τT = F 1
T = F(b). (3.2) Theorems 2.3, 2.4, 3.1 and formula (3.2) imply the following result.
Theorem 3.2 (a) The following lower and upper estimates hold for the function C(T ):
for all T > 0
F(b) − (b) ≤ C(T ) ≤ F(b), (3.3) 1
where b = T .
(b) For large T (i.e. b is small) log T − 1 − C + O log2 T
T = B(T ) ≤ C(T ) ≤ C ∗ (T ) = log T − C + O log2 T
,
T
(3.4) which implies for large T
C ∗ (T ) − 1 + O log2 T
T ≤ C(T ) ≤ C ∗ (T ). (3.5) (c) For small T (i.e. b is large)
T
+ O(T 2 ) ≤ C(T ) ≤ T + O(T 2 ),
2 (3.6) which implies for small T
C ∗ (T )
+ O(T 2 ) ≤ C(T ) ≤ C ∗ (T ).
2
2. The rest of this section is the proofs of Theorems 3.1 and 3.2.
Proof of Theorem 3.1: Denote Bθ (τ) = E θ (τ − θ  τ ≥ θ). Then
C(T ) = inf sup Bθ (τ).
τ ∈M T θ ≥ 0 Since E ∞ I(θ ≤ τ) = E θ I(θ ≤ τ) for any stopping time τ, then for any θ ≥ 0 and any
τ ∈ MT
sup Bθ (τ) · E ∞ I(θ ≤ τ) ≥ Bθ (τ) · E θ I(θ ≤ τ) = E θ (τ − θ)+ .
˜
˜
θ ≥0 (3.7) Since E ∞ I(θ ≤ τ) = P∞ (τ ≤ θ), by integrating in θ the ﬁrst and the last expressions
in (3.7), we ﬁnd
sup Bθ (τ) · E ∞ τ ≥
˜
˜
θ ≥0 ∞
0 E θ (τ − θ)+ dθ. This article is protected by German copyright law. You may copy and distribute this article for your personal use only. Other use is only allowed with written permission by the copyright holder. Quickest detection Since E ∞ τ = T for all τ ∈ MT , we have for τ ∈ MT
∞ 1
T sup Bθ (τ) ≥
˜
˜
θ ≥0 0 E θ (τ − θ)+ dθ and
C(T ) = inf sup Bθ (τ) ≥ inf
˜
τ ∈M T θ ≥ 0
˜ τ ∈M T ∞ 1
T 0 E θ (τ − θ)+ dθ = B(T ), i.e. the left inequality in (3.1) is proved.
∗
Now we shall prove the right inequality in (3.1). Consider the stopping time τT deﬁned
∗
in (1.15). Since τT ∈ MT ,
∗
∗
C(T ) ≤ sup E θ (τT − θ  τT ≥ θ).
θ ≥0 Under the measure P0 , the Markov diffusion process (ψt )t ≥0 satisﬁes (1.14) with
dX t = µ dt + σ dBt , X 0 = 0. (3.8) ∗
For A ≥ 0 we consider stopping times τ A = inf {t ≥ A : ψt ≥ A}. Then for each
s ≥ 0 we have Es ∗
∗
τA − s I τA ≥ s = Es ∗
∗
τ A ◦ θs I τ A ≥ s , ∗
∗
where θs is the “right shift by s” operator, i.e. (τ A ◦ θs )(ω) = τ A (θs ω) and θs ω means the
shifted trajectory (ωu +s )u ≥0 for ω = (ωu )u ≥0 .
(x)
We denote by P0 the distribution of the process (ψt )t ≥0 under the assumptions that
(x)
a disorder takes place at time θ = 0 and ψ0 = x , where x ≥ 0. We also denote by E 0
(x)
(0)
the expectation operator with respect to the measure P0 ; E 0 = E 0 .
∗
∗
Since {τ A ≥ s} = {τ A ≥ s, ψs ≤ A}, by the Markov property of the homogeneous
process (ψt )t ≥0 and (3.8), we ﬁnd that for s ≥ 0
∗
∗
E s (τ A − s  τ A ≥ s) = A
0
A =
0 A =
0 ∗
∗
∗
E s τ A ◦ θs  τ A , ψs = x Ps ψs ∈ dx  τ A ≥ s
(
∗
∗
E 0x)τ A P∞ ψs ∈ dx  τ A ≥ s
(0) ∗
E 0 τ A− x ∗
P∞ ψs ∈ dx  τ A ≥ s ∗
≤ E0 τ A ,
(0) ∗
∗
where τ A−x ≤ τ A P0  a. s. for any x ∈ [0, A]. Hence,
∗
∗
∗
sup E θ (τT − θ  τT ≥ θ) = E 0 τT
θ ≥0 This article is protected by German copyright law. You may copy and distribute this article for your personal use only. Other use is only allowed with written permission by the copyright holder. Feinberg  Shiryaev 458 459 and
∗
∗
∗
C(T ) = inf sup E θ (τ − θ  τ ≥ θ) ≤ sup E θ (τT − θ  τT ≥ θ) = E 0 τT = C ∗ (T ).
τ ∈M T θ ≥ 0 θ ≥0 3. Recall the function F deﬁned in (2.10).
Lemma 3.3 Let ρ = 1. The function
(
∗
V(x ) = E 0x)τT has for all 0 ≤ x ≤ T the representation
V(x ) = F 1
T 1
.
x −F (3.9) In particular,
V(0) = F 1
T . (
Proof: With respect to the measure P0 x) , the Markov diffusion process (ψt )t ≥0 is deﬁned
(x) ∗
by (1.14) with ψ0 = x and (3.8). Hence, for 0 < x < T the function V(x ) = E 0 τT
satisﬁes Kolmogorov’s backward equation (1 + 2ρx )V + ρx 2 V = −1. (3.10)
(x) (x) ∗
∗
In addition, V(x ) = 0 for x ≥ T. Also, it is easy to see that E 0 τT ≤ E ∞ τT .
(x) ∗
(x)
Let us show that E ∞ τT = T − x . Indeed, with respect to the measure P∞ ,
t ψt = x + t + ρ ψs dBs . (3.11) 0
∗
Since 0 ≤ ψs ≤ T for all s ≤ τT , we ﬁnd by taking the expectation in (3.11) with
(x)
(x) ∗
respect to the measure P∞ that E ∞ τT = T − x .
(
(x ∗
∗
Thus, we have that E 0x)τT ≤ E ∞)τT = T − x . This implies that V(x ) is bounded and
we must consider only bounded on [0, T ] solutions of the equation (3.10). It is easy to
check that all such solutions have the following form:
x 1 V(x ) = (C1 − x ) + e ρx
0 where C1 is a constant. − ρ1
u e du , This article is protected by German copyright law. You may copy and distribute this article for your personal use only. Other use is only allowed with written permission by the copyright holder. Quickest detection The condition V(T ) = 0 leads to
x 1 V(x ) = T − x + e ρx 1 1 0 By setting y = 1
ρu , T e− ρu du − e ρT e− ρu du .
1 (3.12) 0 we get
x ∞ 1
ρ e− ρu du =
1 0 1
ρx e−u
du .
u2 (3.13) In addition,
∞
a e−u
du = −
u2 ∞
a 1
e−u d( u ) = e−a
−
a ∞
a e−u
e−a
du =
− (− Ei(−a)). (3.14)
u
a When ρ = 1, formulae (3.12)–(3.14) imply (3.9).
Proof of Theorem 3.2: The inequalities (3.3) follow directly from (3.1), (3.2), and
Theorem 2.3. The inequalities (3.4) and (3.6) follow from the asymptotics (2.19) for the
1
1
function B(T ) = F T −
T. 4 Conditionalextremal problem
1. The conditionalextremal problem is to ﬁnd, for a given T > 0, a stopping time
τ ∗ ∈ MT , if it exists, such that
τ∗ E∞ 0 ψs ds = inf E ∞
τ ∈M T τ ψs ds. 0 Recall that according to Lemma 2.1
inf τ ∈M T τ 1
E∞
T ψs ds = B(T ), 0 ∞ 1
where B(T ) = inf τ ∈MT T 0 E θ (τ − θ)+ dθ.
To solve this problem, we use the traditional “method of Lagrange multipliers.” In
our case this means that we should solve ﬁrst the following extremal problem: ﬁnd
τ inf E ∞ ψs ds − cτ , (4.1) 0 where the inﬁmum is taken over the class of all stopping times τ ≥ 0 with E ∞ τ < ∞.
The constant c > 0, called a “Lagrange multiplier,” can be interpreted as the observation
cost per unit time.
2. With respect to the measure P∞ , the process (ψt )t ≥0 satisﬁes the stochastic differential equation (1.17). This process possesses the following properties: (i) it is a This article is protected by German copyright law. You may copy and distribute this article for your personal use only. Other use is only allowed with written permission by the copyright holder. Feinberg  Shiryaev 460 461 nonnegative submartingale, and (ii) it is a Markov diffusion process. To solve (4.1) we
use a “Markovian approach;” see [19, 13] for details on the Markovian approach in the
optimal stopping theory.
(x
Let P∞) be a distribution of the process (ψt )t ≥0 deﬁned by the stochastic differen(x
tial (1.17) with the initial condition ψ0 = x . Let E ∞) be the expectation operator with
respect to this measure. Denote
∗
(x
Sc (x ) = inf E ∞) τ (ψs − c) ds, (4.2) 0 (x
where inf is taken over all τ ∈ M with E ∞)τ < ∞. According to the general theory of
optimal stopping [19, 13], there exists an optimal stopping rule for the problem (4.1) and
this optimal stopping rule τ ∗ (c) has the form τ ∗ (c) = inf {t ≥ 0 : ψt ≥ x ∗ (c)}, (4.3) where x ∗ (c) is a nonnegative number. The observations should be continued when ψt ∈
[0, x ∗ (c)), and the observations should be stopped when ψt ∈ [x ∗ (c), ∞). Moreover,
∗
L ∞ Sc (x ) = −x + c, x ∈ [0, x ∗ (c)), d
d
∗
with L ∞ = dx + ρx 2 dx 2 and Sc (x ) = 0 if x ≥ x ∗ (c).
∗ ( x ) ≤ 0, since the process can be stopped at time 0. Since (ψ − c) < 0
It is clear that Sc
s
when ψs < c, it is easy to see from (4.2) that
2 x ∗ (c) ≥ c. (4.4) Asymptotic values of x ∗ (c) for c → 0 and for c → ∞ are presented in Lemma 4.6.
According to the Markovian approach in the theory of optimal stopping [19, 13], to
∗
ﬁnd the function Sc (x ), it is necessary to solve the following Stefan problem with an
unknown boundary x (c):
L ∞ Sc (x ) = −x + c,
Sc (x ) = 0,
Sc (x ) = 0, x < x (c),
x ≥ x (c),
x = x (c). (4.5) Let us underline that the unknown elements in (4.5) are the function Sc (x ) and the
boundary point x (c).
Bounded solutions of the equation L ∞ Sc (x ) = −x + c have the form (we consider
ρ = 1 for simplicity)
S(x ) = C1 + cx − G 1
,
x where G is deﬁned in (2.11). The conditions Sc (x (c)) = 0 and Sc (x (c)) = 0 provide
a possibility to ﬁnd the constant C1 and the value of x (c) by solving
F 1
x (c) =c (4.6) This article is protected by German copyright law. You may copy and distribute this article for your personal use only. Other use is only allowed with written permission by the copyright holder. Quickest detection and
1
x (c) C1 = G − cx (c). Therefore, the bounded solution Sc (x ) of (4.5) has the following structure:
Sc (x ) = G 1
x (c) −G 1
x + c(x − x (c)). (4.7) In particular,
Sc (0) = G 1
− cx (c).
x (c) ∗
The solutions Sc (x ) and x (c) from (4.7) and (4.6) are exactly the optimal value Sc (x )
∗ (c) for the optimal stopping problem (4.1) respectively. This
and the optimal boundary x
follows from the general theory of optimal stopping for Markov processes [19, 13]. The
optimality of the stopping time τ ∗ (c) deﬁned in (4.3) with x ∗ (c) such that F( x ∗1c) ) = c can
(
also be established without using the general theory. In fact, the optimality of this stopping
time can be proved by using the obtained solution of the free boundary problem (4.5)
and by applying the “veriﬁcation theorems” and Itˆ ’s formula in the same way as in [21,
o
Chapter 7, Section 2a] or in [13, Theorem 22.1 in Chapter 6]. Thus, we have the following
result. Theorem 4.1 Consider the optimal stopping problem (4.1) for ρ = 1. Then the optimal
value function is
S∗ ( x ) = G 1
x ∗ (c) −G 1
x + c(x − x ∗ (c)), where x ∗ (c) is a unique root of the equation
F 1
x ∗ (c) = c. (4.8) The optimal stopping time is
τ ∗ (c) = inf {t ≥ 0 : ψt ≥ x ∗ (c)}.
3. Theorem 4.1 implies that for each c > 0 the stopping time τ ∗ (c) is optimal in the
(0
class M. This means that for any stopping time τ ∈ M with E ∞) τ < ∞
(0
E ∞) τ ∗ (c)
0 (0
(ψs − c) ds ≤ E ∞) τ (ψs − c) ds. (4.9) 0 1
For a given constant T > 0, let’s set now c = c∗ , where c∗ = F T . Equation (4.8)
implies that x ∗ (c∗ ) = T. Therefore, τ ∗ (c∗ ) ∈ MT and (4.9) yields that for any τ ∈ M This article is protected by German copyright law. You may copy and distribute this article for your personal use only. Other use is only allowed with written permission by the copyright holder. Feinberg  Shiryaev 462 463 (0) with E ∞ τ < ∞
(0
E ∞) τ ∗ (c∗ )
0 (0
ψs ds ≤ E ∞)
τ =
0 τ
0 (0
(0
ψs ds + c∗ E ∞)τ ∗ (c∗ ) − E ∞)τ (0
ψs ds + c∗ T − E ∞) τ . (4.10) (0) If T ≤ E ∞ τ < ∞, i.e. τ ∈ MT , then (4.10) implies
(0
E ∞) τ ∗ (c∗ )
0 (0
ψs ds ≤ E ∞) τ ψs ds. (4.11) 0 Thus, we have the following corollary from Theorem 4.1.
Corollary 4.2 Let c∗ = F 1
T . The stopping time τ ∗ (c∗ ) ∈ MT is optimal within the
(0) class MT , i.e. the inequality (4.11) holds for any τ with T ≤ E ∞ τ < ∞.
Corollary 4.2 and (2.5) imply the following statement.
Corollary 4.3 For any T > 0
∞ inf τ ∈M T 0 E θ (τ − θ)+ dθ = inf τ ∈M T ∞
0 E θ (τ − θ)+ dθ. 4. According to Theorem 3.1,
B(T ) ≤ C(T ) ≤ C ∗ (T ), (4.12) ∗
where C(T ) = inf τ ∈MT supθ ≥0 E θ (τ − θ  τ ≥ θ) and C ∗ (T ) = E 0 τT . The following
theorem states the similar inequalities for the class MT . Theorem 4.4 For any T > 0
B(T ) ≤ C (T ) ≤ C ∗ (T ), (4.13) where C (T ) = inf τ ∈MT supθ ≥0 E θ (τ − θ  τ ≥ θ).
Proof: The right inequality in (4.13) follows from the right inequality in (4.12) and from
MT ⊃ MT . For the proof of the left inequality in (4.13), note that for any stopping time τ
sup E θ (τ − θ  τ ≥ θ) · E ∞ τ =
θ ≥0 ∞
0
∞ ≥
0 =
0 ∞ sup E θ ∗ (τ − θ ∗  τ ≥ θ ∗ ) Pθ (τ > θ) dθ θ ∗ ≥0 E θ (τ − θ  τ ≥ θ) Pθ (τ > θ) dθ
E θ (τ − θ)+ dθ = E ∞ τ
0 ψu du , This article is protected by German copyright law. You may copy and distribute this article for your personal use only. Other use is only allowed with written permission by the copyright holder. Quickest detection where the last equality follows from (2.4). Therefore, for T > 0
inf sup E θ (τ − θ  τ ≥ θ) ≥ inf τ ∈M T θ ≥ 0 Since τ ∈ MT =
inf τ ∈M T E∞ E∞ τ ∈M T τ
0 ψs ds
.
E∞τ (4.14) a≥0 MT +a , τ
0 τ ψs ds
E ∞ 0 ψs ds
= inf inf
= inf B(T + a) = B(T ),
a≥0 τ ∈ MT +a
a ≥0
E∞τ
E∞τ (4.15) where the second equality in (4.15) follows from (2.6), and the last equality follows from
the monotonicity of the function B stated in Lemma 4.5 below. Hence, (4.14) and (4.15)
imply the left inequality in (4.13).
Lemma 4.5 The function B(T ), T > 0, is increasing.
∞ −u Proof: Consider the function F(x ) = ex x e u du introduced in (2.10). We observe
−x
∞ −t
that F(x ) < 1/x . Indeed, this inequality is equivalent to x e t dt < e x , which can be
∞
rewritten as x x ex −t dt < 1. The last inequality holds because
t
∞
x x x −t
e dt =
t ∞
0 x
e−u du <
x +u ∞ e−u du = 1. 0 According to Theorem 2.3, B(T ) = F(b) − (b), where b = 1/T. To complete the proof,
we shall show that the function F(b) − (b) is decreasing. Indeed, we have
F (b) − ∞ (b) =
b 1
F(u )
du − =
u
b ∞
b F(u )
1
−2
u
u du < 0, where the last inequality follows from F(u ) < 1/u .
5. For large and small values of c > 0, the following statement provides the asymptotics for the threshold x ∗ (c), that deﬁnes by equation (4.3) the optimal stopping time
∗
τ ∗ (c), and for the value of Sc (0) deﬁned in (4.2).
Lemma 4.6 (a) For c → 0,
x ∗ (c) = c + c2 + O(c3 ) (4.16) ∗
Sc (0) = − 1 c2 + O(c3 ).
2 (4.17) x ∗ (c) ∼ ec+C (4.18) and (b) For c → ∞, This article is protected by German copyright law. You may copy and distribute this article for your personal use only. Other use is only allowed with written permission by the copyright holder. Feinberg  Shiryaev 464 465 and
∗
Sc (0) ∼ −ec+C , (4.19) where C = 0.577 . . . is Euler’s constant.
Proof: (a) According to (4.8), the threshold x ∗ (c) is the solution of the equation
F(1/x ∗ (c)) = c. Let b = 1/x ∗ (c). Then (2.22) implies that b → ∞ when c → 0.
Therefore x ∗ (c) → 0 and, in view of (2.10) and (2.21),
F(b) = eb (− Ei(−b)) = 1
b − 1
b2 +O 1
b3 . (4.20) Thus for small c > 0
c = x ∗ (c) − (x ∗ (c))2 + O [x ∗ (c)]3
or equivalently
x ∗ (c) = c + (x ∗ (c))2 + O [x ∗ (c)]3 (4.21) and therefore
(x ∗ (c))2 = c2 + O [x ∗ (c)]3 . (4.22) Equations (4.21) and (4.22) imply (4.16).
∗
To ﬁnd the asymptotic for Sc (0) when c → 0, we notice from Theorem 4.1 that
1 ∗
Sc (0) = G x ∗ (c) − cx ∗ (c), (4.23) where
∞ G (b) =
b F(u )
du .
u2 The last equation and (4.20) imply that for large b
G (b) = 1
1
+O 3 .
2b2
b From (4.23), (4.24), and (4.16), we have
∗
Sc (0) = 1∗
2
2 [ x (c)] + O [x ∗ (c)]3 − cx ∗ (c) = 1 (c + c2 )2 − c(c + c2 )2 + O(c3 ) = − 1 c2 + O(c3 ).
2
2 (4.24) This article is protected by German copyright law. You may copy and distribute this article for your personal use only. Other use is only allowed with written permission by the copyright holder. Quickest detection (b) Let c → ∞. Then the equation F(1/x ∗ (c)) = c and (2.20), (2.22), and (2.32)
imply that x ∗ (c) → ∞ and
log x ∗ (c) − C + O log x ∗ (c)
x ∗ (c) = c. (4.25) Thus for c → ∞
x ∗ (c) ∼ ec+C . (4.26) ∗
To ﬁnd the asymptotic of Sc (0) for c → ∞, we rewrite (4.23) as
∗
Sc (0) = x ∗ (c) 1
1
G∗
x ∗ (c)
x (c) As follows from (2.7), (2.8), and (2.13), B(T ) =
large T −c . 1
1
TG T B(T ) = log T − 1 − C + O (4.27) and, according to (2.19), for log2 T
.
T Hence for large x ∗ (c)
1
1
G∗
x ∗ (c)
x (c) = log x ∗ (c) − 1 − C + O log2 x ∗ (c)
x ∗ (c) and (4.27) and (4.25) imply
log2 x ∗ (c)
x ∗ (c) ∗
Sc (0) = x ∗ (c) log x ∗ (c) − 1 − C + O = x ∗ (c) −1 + O log2 x ∗ (c)
x ∗ (c) . −c
(4.28) Formulae (4.28) and (4.26) imply (4.19).
6. Consider asymptotics for x ∗ (c) found in Lemma 4.6.
Remark 4.7 Formula (4.16) shows that x ∗ (c) insigniﬁcantly exceeds c when c is small:
x ∗ (c) − c ∼ c2 . However, if c is large, x ∗ (c) ∼ ec+C . At the ﬁrst glance, this result seems
strange because, when the values of ψs are close to x ∗ (c), the values of ψs − c are large
and their contributions to the objective function are also positive, while it appears that for
an optimal policy these contributions should be negative. In fact, there is no contradiction
here, since the process (ψt )t ≥0 is positive recurrent with respect to the measure P∞ , i.e.
(x)
E ∞ σx < ∞ for any for any x > 0, where σx = inf {t > 0 : ψt = x }. For the ﬁrst time,
this was pointed out by Pollak and Siegmund [14], where it was also noticed that this
(x)
process has an invariant distribution F( y) = limt →∞ P∞ (ψt ≤ y), y > 0, for any initial
state x ≥ 0. This article is protected by German copyright law. You may copy and distribute this article for your personal use only. Other use is only allowed with written permission by the copyright holder. Feinberg  Shiryaev 466 467 To ﬁnd this distribution, we write Kolmogorov’s forward equation for the density
(x)
f (t, y) of the distribution F(t, y) = P∞ (ψt ≤ y),
∂f
∂f
∂2
=−
+ ρ 2 ( y2 f ),
∂t
∂y
∂y y > 0. Therefore, the density f = f ( y) of the invariant distribution F = F( y) satisﬁes the
equation
df
d2
= ρ 2 ( y2 f ),
dy
dy y > 0, whose nonnegative solution with the condition F(0+) = 0 (which follows from positivity
a.s. of ψt for all t > 0) and natural condition F(∞) = 1, as it is easy to ﬁnd, has the form
f ( y) = 1 −1/(ρ y)
e
,
y2 y > 0; see [15, 2.1.2.129 and 2.1.2.103]). Therefore the invariant distribution F = F( y) is given
by the formula
F( y) = e−1/(ρ y), y > 0. This is the Fr´ chettype distribution, which is well known in the theory of extreme value
e
distributions [7]. 5 Comparison of two minimax variants
1. Recall that in Variant (C) the value of the criterion is deﬁned as
C(T ) = inf sup E θ (τ − θ  τ ≥ θ).
τ ∈M T θ ≥ 0 In the statistical literature there are many investigations of another minimax criterion
presented in Variant (D).
∗
Variant (D). In the class MT = {τ ≥ 0 : E ∞ τ = T } ﬁnd a stopping time σT , if it
exists, such that
∗
sup ess sup E θ ((σT − θ)+  Fθ )(ω) = inf sup ess sup E θ ((τ − θ)+  Fθ )(ω). (5.1)
θ ≥0 ω τ ∈M T θ ≥ 0 ω Taking ess sup of the conditional expectations E θ (·  Fθ )(ω) essentially means that we
optimize for the worst possible situation at the time θ when the disorder happens.
For discrete time, criterion (5.1) was introduced by Lorden [10] who proved that
the socalled CUSUM method of Page [12] is asymptotically optimal for this criterion
when T → ∞. Later Moustakides [11] and Ritov [16] proved that the CUSUM method This article is protected by German copyright law. You may copy and distribute this article for your personal use only. Other use is only allowed with written permission by the copyright holder. Quickest detection is indeed optimal. The continuous time model (1.1) was investigated by Beibel [1] and
Shiryaev [20]. They proved that the exponential CUSUM process
γt = sup
θ ≤t Lt
,
Lθ t ≥ 0, Lt
∗
(cf. ψt = 0 L θ dθ ) is the sufﬁcient statistics and the stopping time σT = inf {t ≥ 0 :
γt ≥ D}, where D is the root of D − 1 − log D = T , is optimal in the class MT ; we recall
that here ρ = 1.
Denote
t D(T ) = inf sup ess sup E θ ((τ − θ)+ Fθ )(ω)
τ ∈M T θ ≥ 0 ω and recall that
1∞
E θ (τ − θ)+ dθ,
τ ∈M T T 0
C(T ) = inf sup E θ (τ − θ  τ ≥ θ),
B(T ) = inf τ ∈M T θ ≥ 0 ∗ ∗
∗
∗
C (T ) = sup E θ τT − θ  τT ≥ θ = E 0 τT .
θ ≥0 The following inequalities summarize for any ρ > 0 the relationship between B(T ),
C(T ), C ∗ (T ), and D(T ):
B(T ) ≤ C(T ) ≤ C ∗ (T ),
B(T ) ≤ C(T ) ≤ D(T ). (5.2) For large T
B( T ) =
C ∗ (T ) =
D(T ) = 1
log2 ρT
log(ρT ) − 1 − C + O
ρ
ρT
1
log2 ρT
log(ρT ) − C + O
ρ
ρT
1
1
log(ρT ) − 1 + O
ρ
ρT , ,
(5.3) , where C = 0.577 . . . is Euler’s constant; see (2.19) for B(T ), (3.4) and (3.6) for C(T )
and C ∗ (T ), and [1, 20] for D(T ).
The inequalities (5.2) and asymptotics (5.3) imply that
C ∗ (T ) − 1
log ρT
+O
ρ
ρT ≤ C(T ) ≤ C ∗ (T ), ∗
which shows that the stopping time τT is asymptotically optimal for Variant (C). Note
∗ ( T ) for large T. As follows from the deﬁnitions of C( T ) and D( T ), it is
that D(T ) ≤ C This article is protected by German copyright law. You may copy and distribute this article for your personal use only. Other use is only allowed with written permission by the copyright holder. Feinberg  Shiryaev 468 469 always true that C(T ) ≤ D(T ). So, for large T we have not only the inequality
C( T ) ≤ 1
log2 ρT
log(ρT ) − C + O
ρ
ρT , but a slightly better inequality
C( T ) ≤ 1
log2 ρT
log(ρT ) − 1 + O
ρ
ρT follows from (5.2) and (5.3). In connection to formulas (5.3), it is useful to remark that ρ
1
has the dimension sec and ρT is dimension free. Acknowledgements. This research was partially supported by NSF (National Science
Foundation) Grants DMI0300121 and DMI0600538, by a grant from NYSTAR (New
York State Ofﬁce of Science, Technology, and Academic Research), and by RFBR (Russian Foundation for Basic Research), Grant 050100944a. References
[1] M. Beibel. A note on Ritov’s Bayes approach to the minimax property of the
CUSUM procedure. Annals of Statistics, 24:1804–1812, 1996.
[2] E. Carlstein, H.G. M¨ ller, and D. Siegmund, editors. ChangePoint Problems.
u
Institute of Mathematical Statistics, IMS Lecture Notes Monograph Series 23, 1994.
[3] E. B. Dynkin. Markov Processes. Volumes I, II. SpringerVerlag, 1965.
[4] A. Erd´ lyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi. Tables of Integral
e
Transforms. Volume I. McGrawHill, 1954.
[5] I. S. Gradshteyn and I. M. Ryzhik. Tables of Integrals, Series, and Products. Academic Press, 1994.
[6] D. M. Hawkins and D. H. Olwell. Cumulative Sum Charts and Charting for Quality
Improvement. Springer, 1998.
[7] S. Kotz and S. Nadarajah. Extreme Value Distributions. Imperial College Press,
2000.
[8] N. N. Lebedev. Special Functions and Their Applications. PrenticeHall, 1965.
[9] R. Sh. Liptser and A. N. Shiryaev. Statistics of Random Processes. I. General
Theory; II. Applications. SpringerVerlag, 1977, 1978.
[10] G. Lorden. Procedures for reacting to a change in distribution. Annals of Mathematical Statistics, 42:1897–1908, 1971. This article is protected by German copyright law. You may copy and distribute this article for your personal use only. Other use is only allowed with written permission by the copyright holder. Quickest detection Feinberg  Shiryaev [11] G. V. Moustakides. Optimal stopping times for detecting changes in distributions.
Annals of Statistics, 14:1379–1387, 1986.
[12] E. S. Page. Continuous inspection schemes. Biometrika, 41:100–115, 1954.
[13] G. Peskir and A. N. Shiryaev. Optimal Stopping and FreeBoundary Problems.
Birkh¨ user, 2006.
a
[14] M. Pollak and D. Siegmund. A diffusion process and its applications to detecting
a change in the drift of Brownian motion. Biometrika, 72:267–280, 1985.
[15] A. D. Polyanin and V. F. Zaitsev. Handbook of Exact Solutions for Ordinary Differential Equations. CRC Press, 1995.
[16] Y. Ritov. Decision theoretic optimality of the CUSUM procedure. Annals of
Statistics, 18:1464–1469, 1990.
[17] A. N. Shiryaev. The problem of the most rapid detection of a disturbance in a stationary process. Soviet Mathematics, Doklady, 2:795–799, 1961.
[18] A. N. Shiryaev. On optimum methods in quickest detection problems. Theory of
Probability and Its Applications, 8:22–46, 1963.
[19] A. N. Shiryayev [Shiryaev]. Optimal Stopping Rules. SpringerVerlag, 1978.
[20] A. N. Shiryaev. Minimax optimality of the method of cumulative sums (CUSUM)
in the continuous time case. Russian Mathematical Surveys, 51:750–751, 1996.
[21] A. N. Shiryaev. Essentials of Stochastic Finance. World Scientiﬁc, 1999.
[22] A. G. Tartakovsky and V. V. Veeravalli. General asymptotic Bayesian theory of
quickest change detection. Theory of Probability and Its Applications, 49:458–497,
2005.
Eugene A. Feinberg
Department of Applied Mathematics
and Statistics
Stony Brook University
Stony Brook, NY 117943600
USA
Eugene.Feinberg@sunysb.edu Albert N. Shiryaev
Steklov Mathematical Institute
Gubkina Str. 8
Moscow 119991
Russia
albertsh@mi.ras.ru This article is protected by German copyright law. You may copy and distribute this article for your personal use only. Other use is only allowed with written permission by the copyright holder. 470 ...
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