On a Quickest Detection Problem

Abstract

In a recent paper A. N. Shiryaev [1] discusses the problem of detecting the arrival of a "disorder" in an observed stochastic process, as quickly as possible subject to a limitation on the number of false alarms. He considers two versions of a simple model. In the first, the disorder arrives at a discrete instant $\theta$ according to a geometric distribution. The process disturbed by this event consists of a sequence of independent observations $\{\xi_t\}$, such that $\xi_1, \xi_2, \cdots, \xi_{\theta - 1}$ arise from a certain distribution $F_0$, whereas $\xi_\theta, \xi_{\theta + 1}, \cdots$ come from a different distribution $F_1$. In the continuous time version of the model, the a-priori distribution of $\theta$ is exponential: $P(\theta > t) = e^{-\lambda t} \quad (t \geqq 0)$ and the disorder is represented by a change in the mean drift of an observed Wiener process $\{\eta(t)\}$. More precisely, for any given value of $\theta$, this process has independent normal increments $\delta\eta = \eta(t + \delta t) - \eta(t)$, with \begin{align*}E (\delta\eta) = 0 \quad (0 \leqq t t + s \mid \theta > s) = P(\theta > t \mid \theta > 0).$ He deduces that the optimal policy for the period following any false alarm must correspond exactly with the procedure applied initially, before the first inspection. In addition, he proves the existence of a critical level $p^\ast = p^\ast(N)$ such that, in general, observations should continue so long as $0 \leqq p(t) 0$, not depending on its outcome, whereas any delay in detecting the arrival of the disorder leads to a cost $c > 0$ per unit time. Hence the total expected cost is $K(N + 1) + c\tau$, which depends both on the decision procedure and on the initial condition $p(0) = 0$. The minimization will be based on the calculation of the minimum expected future cost $f_\ast(\varphi)$, as a function of the current state $\varphi(t) = p(t)/(1 - p(t))$, by solving a certain differential equation with special boundary conditions. A heuristic argument, in which one simply assumes that $f_\ast(\varphi)$ is suitably differentiable, can be given without much difficulty. But, strictly speaking, it is not clear that risk functions such as $f_\ast(\operatorname{var})$, each of whose values is defined as the infimum of a class of expectations, are sufficiently well behaved. This difficulty is often encountered in statistical applications of dynamic programming to processes in continuous time. Typically, it is extremely difficult to establish the required differentiability properties directly and it is necessary to seek an indirect justification by means of existence and uniqueness theorems. In our case the formal solution can be produced explicitly and, because of this, its justification is much easier. Nevertheless, the approach is complicated by the need to establish several preliminary results, and the discussion of these special properties is limited to a brief indication of the main steps, in the hope that the essential structure of the argument will be more generally useful. Section 2 is concerned with the information process $\{\varphi(t)\}$. Its relation to the observed process $\{\eta(t)\}$ is described and certain properties of its increments are collected for later use. The main argument begins in Section 3 with a discussion of sub-optimal decision procedures defined by specifying an open continuation set $\mathscr{C}$ within the space $\lbrack0, \infty)$ of possible "initial" states $\varphi$. It is shown that any risk function $f(\varphi)$ can be determined for each sub-interval of the corresponding set $\mathscr{C}$ by solving the basic differential equation appropriately. Sections 4 and 5 consider the special solution $f_\ast(\varphi)$ which represents the optimal decision procedure, and give the required verification that $f_\ast(\varphi)$ is uniformly minimal. The final section contains a brief analysis of the operating characteristics of the optimal policy and indicates the importance of evaluating the particular minimum risk $f_\ast(0)$.