sd0604_445a - Quickest detection of drift change for...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 so-called 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 right-hand 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 two-sided inequalities (3.5) for the value function on the right-hand 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 conditional-extremal 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 continuous-time 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 Conditional-extremal problem 1. The conditional-extremal 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´ chet-type 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 so-called 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 DMI-0300121 and DMI-0600538, by a grant from NYSTAR (New York State Office of Science, Technology, and Academic Research), and by RFBR (Russian Foundation for Basic Research), Grant 05-01-00944-a. 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. Change-Point Problems. u Institute of Mathematical Statistics, IMS Lecture Notes Monograph Series 23, 1994. [3] E. B. Dynkin. Markov Processes. Volumes I, II. Springer-Verlag, 1965. [4] A. Erd´ lyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi. Tables of Integral e Transforms. Volume I. McGraw-Hill, 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. Prentice-Hall, 1965. [9] R. Sh. Liptser and A. N. Shiryaev. Statistics of Random Processes. I. General Theory; II. Applications. Springer-Verlag, 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 Free-Boundary 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. Springer-Verlag, 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 11794-3600 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 ...
View Full Document

Ask a homework question - tutors are online