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A remark on the quickest detection problems
... Jump distribution given by (28) together with Theorem 3 allows us to formulate the following lemma. ...
... Assume that jump distribution F ∞ of the process X ∞ is given by (28) and jump intensity is equal to µ ∞ . Assume also that there exists a vector z r0 = (z r0,1 , . . . ...
... Considering the jump distributions F ∞ and F r0 given by (28) and (29), the system (20) consists of equations ...
In this paper, using Bayesian approach, we solve the quickest drift change detection problem for a multidimensional L\'evy process consisting of both a continuous gaussian part and a jump component. We allow a general a priori distribution of the change point as well as random post-change drift parameter. Classically, our optimality criterion is based on a probability of false alarm and an expected delay of the detection. The main technique uses the optimal stopping theory and is based on solving a certain free-boundary value problem. The paper is supplemented by an extensive numerical analysis related with the construction of the Generalized Shiryaev-Roberts statistic applied to analysis of Polish life tables (after proper calibration) and predicting the joint drift change in the correlated force of mortality of men and women.
... The optimal stopping time τ * obtained in Theorem 1 is also optimal for minimizing E x | τ − θ | instead of V * ( x ) given in Problem 1 , for c = λ. It follows directly from Shiryaev [24] , because (40) in the proof of Lemma 2 in the Appendix for details). ...
... Remark 2.4. From the remark by Shiryaev [32], it follows that ...
We consider a continuously observable process, which behaves like a standard Brownian motion up to a random time τ1 and as a Brownian motion with a known drift after τ1. At a stopping-time τ2 that happens after the time τ1, an observable event occurs. We address the problem of detecting the change in the system's behaviour prior to the occurrence of the observable event. In particular, our formulation takes into account the information provided by the non-occurrence of the observable event and where it is favourable to “raise the alarm” before this event. We show that this problem can be reduced to a one-dimensional optimal stopping problem, to which we derive an explicit solution.
... (ii) following from (16,17,19)), the optimal thresholds in (49) become ξ 1 = ξ 2 = . . ...
In the compound Poisson disorder problem, arrival rate and/or jump distribution of some compound Poisson process changes suddenly at some unknown and unobservable time. The problem is to detect the change (or disorder) time as quickly as possible. A sudden regime shift may require some countermeasures be taken promptly, and a quickest detection rule can help with those efforts. We describe complete solution of the compound Poisson disorder problem with several standard Bayesian risk measures. Solution methods are feasible for numerical implementation and are illustrated by examples.
In this paper we give a solution to the quickest drift change detection problem for a multivariate Lévy process consisting of both continuous (Gaussian) and jump components in the Bayesian approach. We do it for a general 0-modified continuous prior distribution of the change point as well as for a random post-change drift parameter. Classically, our criterion of optimality is based on a probability of false alarm and an expected delay of the detection, which is then reformulated in terms of a posterior probability of the change point. We find a generator of the posterior probability, which in case of general prior distribution is inhomogeneous in time. The main solving technique uses the optimal stopping theory and is based on solving a certain free-boundary problem. We also construct a Generelized Shiryaev-Roberts statistic, which can be used for applications. The paper is supplemented by two examples, one of which is further used to analyze Polish life tables (after proper calibration) and detect the drift change in the correlated force of mortality of men and women jointly.
We study the Wiener disorder detection problem where each observation is associated with a positive cost. In this setting, a strategy is a pair consisting of a sequence of observation times and a stopping time corresponding to the declaration of disorder. We characterize the minimal cost of the disorder problem with costly observations as the unique fixed point of a certain jump operator, and we determine the optimal strategy.
The paper deals with the quickest detection of the disorder of a Brownian motion on a finite interval. The unknown moment of disorder is assumed to be uniformly distributed on the interval. The Bayesian and absolute criteria are used as optimality tests. The problem is reduced to a classic optimal stopping problem where the optimal stopping time may be obtained as a solution to integral equations. The existence and uniqueness of the solution of integral equations are proved analytically.
Given a standard Brownian motion Bμ = (Btμ)0≤t≤1 with drift μ ∈ IR, letting S tμ = max0≤s≤t Bsμ for t ∈ [0, 1], and denoting by θ the time at which S1μ is attained, we consider the optimal prediction problem V* = inf0≤τ≤1 E|θ - τ | where the infimum is taken over all stopping times τ of Bμ. Reducing the optimal prediction problem to a parabolic free-boundary problem and making use of local time-space calculus techniques, we show that the following stopping time is optimal: τ* = inf { 0 ≤ t ≤ 1| Stμ - Btμ ≥ b(t)} where b : [0, 1] → ℝ is a continuous decreasing function with b(1) = 0 that is characterized as the unique solution to a nonlinear Volterra integral equation. This also yields an explicit formula for V* in terms of b. If μ = 0 then there is a closed form expression for b. This problem was solved in [14] and [4] using the method of time change. The latter method cannot be extended to the case when μ ≠ 0 and the present paper settles the remaining cases using a different approach. It is also shown that the shape of the optimal stopping set remains preserved for all Lévy processes.
The article contains the author’s answers to comments of the discussants on his article [ibid. 29, No. 4, 345–385 (2010; Zbl 1203.62137)]. In addition, some new results are included.
A process X is observed continuously in time; it behaves like Brownian motion with drift, which changes from zero to a known constant ϑ>0 at some time τ that is not directly observable. It is important to detect this change when it happens, and we attempt to do so by selecting a stopping rule T* that minimizes the “expected miss” E|T−τ| over all stopping rules T. Assuming that τ has an exponential distribution with known parameter λ>0 and is independent of the driving Brownian motion, we show that the optimal rule T* is to declare that the change has occurred, at the first time t for which
. Here, with Λ=2λ/ϑ², the constant p* is uniquely determined in (½,1) by the equation
.
In this paper optimum methods are developed for observing a process (1), in which the moment when a “disorder” $\theta$ appears is not known. The basic quantity characterizing the quality of this observation method is the mean time delay ${\boldsymbol \tau}$ for detection of a disorder.After making assumption (4) it is shown that for a given false alarm probability $\omega$ or for a given ${\bf N}$ — mathematical expectation of false alarm numbers occurring up to the moment the disorder appears — the observation method minimizing ${\boldsymbol \tau } = {\boldsymbol \tau } (\omega)$ or ${\boldsymbol \tau} = {\boldsymbol \tau} ({\boldsymbol N})$ is based on an observation of a posteriors probability (23).In § 3 a case is considered, wherein, the disorder appears on the background of steadystate conditions arising when the disorder is absent. A method is found for minimizing ${\boldsymbol \tau} = {\boldsymbol \tau} ({\bf T})$ for a set ${\bf T}$ — mathematical expectation of the time between two false alarms...
The Poisson disorder problem seeks to determine a stopping time which is as close as possible to the (unknown) time of 'disorder' when the intensity of an observed Poisson process changes from one value to another. Partial answers to this question are known to date only in some special cases, and the main purpose of the present paper is to describe the structure of the solution in the general case. The method of proof consists of reducing the initial (optimal stopping) problem to a free-boundary differential-difference problem. The key point in the solution is reached by specifying when the principle of smooth fit breaks down and gets superseded by the principle of continuous fit. This can be done in probabilistic terms (by describing the sample path behaviour of the a posteriori probability process) and in analytic terms (via the existence of a singularity point of the free-boundary equation).