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Article Excerpt
We solve the Poisson disorder problem when the delay is penalized exponentially. Our objective is to detect as quickly as possible the unobservable time of the change (or disorder) in the intensity of a Poisson process. The disorder time delimits two different regimes in which one employs distinct strategies (e.g., investment, advertising, manufacturing). We seek a stopping rule that minimizes the frequency of false alarms and an exponential (unlike previous formulations, which use a linear) cost function of the detection delay. In the financial applications, the exponential penalty is a more apt measure for the delay cost because of the compounding of the investment growth. The Poisson disorder problem with a linear delay cost was studied by Peskir and Shiryaev [2002. Solving the Poisson Disorder Problem. Advances in Finance and Stochastics. Springer, Berlin, Germany, 295-312], which is a limiting case of ours.
Key words: Poisson disorder problem; quickest detection; optimal stopping; differential-delay equations
MSC2000 subject classification : Primary: 62L 10; secondary: 62L 15, 62C 10, 60G40
OR/MS subject classification: Primary: statistics: Bayesian; estimation; secondary: dynamic programming/optimal control: applications
1. Introduction. In this paper, we address a change-detection problem involving Poisson processes. Suppose that we observe a Poisson process X = {[X.sub.t]; t [greater than or equal to] 0} whose intensity changes from [[lambda].sub.0] to [[lambda].sub.1] at some random time [theta]. The "disorder time" [theta] is unobservable, but has a known a priori probability distribution: it equals zero with probability [pi] [member of] [0, 1) and has exponential distribution with rate [lambda] given that it is positive. The parameters [lambda], [[lambda].sub.0], [[lambda].sub.1] are known positive constants. The Poisson disorder problem is to detect the change-time [theta] by using the observations of X. Here, we are interested in the best detection rule which minimizes the expected sum of the frequency of the false alarms and an exponential cost function of the detection delay.
The change-detection and sequential hypothesis testing problems about the drift of a Wiener process have been extensively studied; see, e.g., Shiryaev [17, 15, Chapter IV], Beibel [3, 2], Karatzas [8], Moustakides [10]. Similar questions for the intensity of a Poisson process also draw significant attention, because Poisson processes are often used to model abrupt changes, such as sudden price movements in stock markets, changes in credit ratings due to defaults, changes in the intensity of earthquakes, product failures in a manufacturing system, etc.
Particularly, the Poisson disorder problem with linear penalty functions of the delay has been investigated well; see, e.g., Galchuk and Rozovsky [7], Davis [6], Peskir and Shiryaev [12, 11]. However, in many applications the cost of the lost opportunity due to the detection delay exponentiates with the delay time. Therefore, it is captured better by an exponential penalty function than by a linear one.
As a simple motivating example from quality control, let us consider an assembly line whose finished products are continuously inspected for defects. A sudden upward shift (e.g., from low [[lambda].sub.0] to high [[lambda].sub.1]) in the rate of the number of defective items (the observation process X) may have an assignable cause and warrant an investigation at some fixed cost. A good control policy should balance the costs of false alarm (due to unnecessary inspection) and detection delay (due to lost production time and raw materials, scrapping or recycling, etc.).
A firm often measures its financial losses and gains by compounding those at its own internal rate of return (IRR). Let us denote our firm's IRR by [alpha]. Typically the production rate of an assembly line is constant, say one item per unit time. Suppose also that a defective item costs one dollar. If the production system goes out of control at time [theta], and an alarm is given at some later time [tau], then every dt units of defective items at each t [member of] [[theta], [tau]) will cost exp{[alpha]([tau] - t)}dt at the detection time [tau]. Therefore, total cost of detection delay equals
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]
is an exponential function of the detection delay time [([tau] - [theta]).sup.+]. In addition, if each false alarm costs 1/(c[alpha]) dollars on average, then an optimal alarm time [tau] should minimize the expected total inspection cost, which is proportional to
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (1)
i.e., the alarm time [tau] should solve optimally some Poisson disorder problem with an exponential penalty for delay.
The only alternative to exponential delay penalty in the literature (see, e.g., Galchuk and Rozovsky [7], Davis [6], Peskir and Shiryaev [12]) is the linear delay penalty, as in
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2)
From decision theoretic point of view, the penalty [??][[([tau] - [theta]).sup.+]] is the choice of a risk-neutral decision maker. Indeed, if the risks implied by the fluctuations in the delay time [([tau] - [theta]).sup.+] are important to a decision maker, then the linear penalty [??][[([tau] - [theta]).sup.+]] falls short.
On the other hand, the exponential penalty [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] not only captures the variability of the delay time, but also reflects the risk-sensitivity of a risk-averse decision maker; see Whittle [18, 19]. By adjusting the parameter [alpha], the decision maker can tune the exponential penalty to his/her risk preferences. To see these, let us replace in (2) the linear penalty [??][[([tau] - [theta]).sup.+]] with the exponential penalty [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and use the identity (see, for example, Bensoussan [5, p. 54])
[e.sup.[alpha]x] = 1 + [alpha]x + [[alpha].sup.2][x.sup.2] [[integral].sup.1.sub.0] [[integral].sup.s.sub.0] [e.sup.r[alpha]x] dr ds, x [greater than or equal to]
to rewrite it. We obtain
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3)
Hence, the exponential penalty on the left accounts for the losses in (2) as well as the effect of the second-order terms in the delay time. For large values of [alpha], every alarm time [tau] causing high variations in the delay time [([tau] - [theta]).sup.+] is now punished severely according to (3). For small values of [alpha], the punishment is lesser, and we retrieve the risk-neutral (linear) case (2) if we let [alpha] go to zero. Hence, the exponential penalty contains the linear penalty as a sub case and allows the risk preferences to be added to the analysis by a natural mechanism.
The importance of exponential delay penalty was recognized first by Poor [13], who solved a quickest-detection problem with exponential delay cost in the discrete-time setting. Later, Beibel [3] solved the Wiener disorder problem with the same cost function. The Poisson disorder problem with an exponential penalty function of the detection delay is studied for the first time in this paper, to the best of our knowledge.
To solve the Poisson disorder problem, we first show that it is equivalent to an optimal stopping problem for a two-dimensional jump process ([PI], [PHI]) in [0, 1] x [[??].sub.+]. For every t [greater than or equal to] 0, [[PI].sub.t], is the conditional probability that the change occurred at or before time t given the past observations of X. On the other hand, [[PHI].sub.t], is essentially the likelihood-ratio process [[PSI].sub.t] [??] [[PI].sub.t]/(1 - [[PI].sub.t]) with an adjustment, adapted to the history of X, reflecting the exponential detection delay cost; see (7). The optimal stopping problem is reduced to a free-boundary problem involving a differential delay equation. By means of a key verification lemma, one solution of the free-boundary problem is identified as the value function of the optimal stopping problem. The optimal stopping rule turns out a threshold type for the process [PHI] regardless of [PI]: declare that the disorder happened at or before time t [greater than or equal to] as soon as [[PHI].sub.t] exceeds a suitable threshold (constant over time).
We characterize the optimal threshold and the value function of the optimal stopping problem. To calculate the threshold and the value function, we describe an efficient numerical method using bisection search on the real line and finite-difference method for differential-delay equations.
Our systematic numerical method also complements the work of Peskir and Shiryaev [12], whose problem is a limiting case of ours. Let us also mention that Beibel [3] reduced the Wiener disorder problem with exponential penalty function to a similar optimal stopping problem and solved it as a generalized parking problem. Beibel's approach relies on the continuity of the paths of the process [PI]. For the Poisson disorder problem, the process [PI] has jumps, and the related optimal stopping problem cannot be formulated as a generalized parking problem.
In the next section, we give a precise description of our problem and formulate the equivalent optimal stopping problem in (6)-(8). The latter is solved, and an optimal Bayes rule and the minimum Bayes cost function are determined in [section] 3; see Propositions 3.1-3.4. To calculate the optimal decision rules, numerical methods are also described; they are illustrated on examples in Figures 2 and 3 in [section] 3. Long proofs are presented in [section] 4 and in the appendix.
[FIGURES 2-3 OMITTED]
Let us remark that our analysis of the free-boundary problem might be useful to solve other quickest-detection problems; see, for example, Bayraktar et al. [1] for an application to "standard" Poisson disorder problems.
2. The problem. Let ([OMEGA], [??], [[??].sub.[pi]]) be a probability space hosting two independent Poisson processes [X.sup.0] = [([X.sup.0.sub.t]).sub.t [greater than or equal to] 0] and [X.sup.1] = [([X.sup.1.sub.t]).sub.t [greater than or equal to] 0] with rates [[lambda].sub.0] and [[lambda].sub.1], respectively, and a random variable [theta], independent of the processes [X.sup.0] and [X.sup.1], having the distribution
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], t [greater than or equal to] 0.
The processes [X.sup.0], [X.sup.1] and the random variable [theta] are unobservable. The observation process is
[X.sub.t] = [[integral].sup.t.sub.0] [1.sub.{s [less than or equal to] [theta]}] d[X.sup.0.sub.s] + [[integral].sup.t.sub.0] [1.sub.{s > [theta]}] d[X.sup.1.sub.s], t [greater than or equal to] 0, (4)
with the natural filtration [[??].sup.X] = [([[??].sup.X.sub.t]).sub.t [greater than or equal to] 0] (modified suitably to make...
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