Semih Onur Sezer

MULTISOURCE BAYESIAN SEQUENTIAL CHANGE DETECTION

Suppose that local characteristics of several independent compound Poisson and Wiener processes change suddenly and simultaneously at some unobservable disorder time. The problem is to detect the disorder time as quickly as possible after it happens and minimize the rate of false alarms at the same time. These problems arise, for example, from managing product quality in manufacturing systems and preventing the spread of infectious diseases. The promptness and accuracy of detection rules improve greatly if multiple independent information sources are available. Earlier work on sequential change detection in continuous time does not provide optimal rules for situations in which several marked count data and continuously changing signals are simultaneously observable. In this paper, optimal Bayesian sequential detection rules are developed for such problems when the marked count data is in the form of independent compound Poisson processes, and the continuously changing signals form a multi-dimensional Wiener process. An auxiliary optimal stopping problem for a jump-diffusion process is solved by transforming it first into a sequence of optimal stopping problems for a pure diffusion by means of a jump operator. This method is new and can be very useful in other applications as well, because it allows the use of the powerful optimal stopping theory for diffusions.

Compound Poisson Disorder Problem

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.

Sequential multi-hypothesis testing for compound Poisson processes

Suppose that there are finitely many simple hypotheses about the unknown arrival rate and mark distribution of a compound Poisson process, and that exactly one of them is correct. The objective is to determine the correct hypothesis with minimal error probability and as soon as possible after the observation of the process starts. This problem is formulated in a Bayesian framework, and its solution is presented. Provably convergent numerical methods and practical near-optimal strategies are described and illustrated on various examples.

Finite Horizon Decision Timing with Partially Observable Poisson Processes

We study decision timing problems on finite horizon with Poissonian information arrivals. In our model, a decision maker wishes to optimally time her action in order to maximize her expected reward. The reward depends on an unobservable Markovian environment, and information about the environment is collected through a (compound) Poisson observation process. Examples of such systems arise in investment timing, reliability theory, Bayesian regime detection and technology adoption models. We solve the problem by studying an optimal stopping problem for a piecewise-deterministic process, which gives the posterior likelihoods of the unobservable environment. Our method lends itself to simple numerical implementation and we present several illustrative numerical examples.

Multisource Bayesian sequential binary hypothesis testing problem

We consider the problem of testing two simple hypotheses about unknown local characteristics of several independent Brownian motions and compound Poisson processes. All of the processes may be observed simultaneously as long as desired before a final choice between hypotheses is made. The objective is to find a decision rule that identifies the correct hypothesis and strikes the optimal balance between the expected costs of sampling and choosing the wrong hypothesis. Previous work on Bayesian sequential hypothesis testing in continuous time provides a solution when the characteristics of these processes are tested separately. However, the decision of an observer can improve greatly if multiple information sources are available both in the form of continuously changing signals (Brownian motions) and marked count data (compound Poisson processes). In this paper, we combine and extend those previous efforts by considering the problem in its multisource setting. We identify a Bayes optimal rule by solving an optimal stopping problem for the likelihood-ratio process. Here, the likelihood-ratio process is a jump-diffusion, and the solution of the optimal stopping problem admits a two-sided stopping region. Therefore, instead of using the variational arguments (and smooth-fit principles) directly, we solve the problem by patching the solutions of a sequence of optimal stopping problems for the pure diffusion part of the likelihood-ratio process. We also provide a numerical algorithm and illustrate it on several examples.

Online Change Detection for a Poisson Process with a Phase-Type Change-Time Prior Distribution

Abstract We consider a change detection problem in which the arrival rate of a Poisson process changes suddenly at some unknown and unobservable disorder time. It is assumed that the prior distribution of the disorder time is known. The objective is to detect the disorder time with an online detection rule (a stopping time) in a way that balances the frequency of false alarms and detection delay. So far in the study of this problem, the prior distribution of the disorder time is taken to be exponential distribution for analytical tractability. Here, we will take the prior distribution to be a phase-type distribution, which is the distribution of the absorption time of a continuous time Markov chain with a finite state space. We find an optimal stopping rule for this general case. We illustrate our findings on two numerical examples.

On the single-leg airline revenue management problem in continuous time

We consider the single-leg airline revenue management problem in continuous time with Poisson arrivals. Earlier work on this problem generally uses the Hamilton–Jacobi–Bellman equation to find an optimal policy whenever the value function is differentiable and is a solution to this equation. In this paper, we employ a different probabilistic approach, which does not rely on the smoothness of the value function. Instead, we use a continuous-time discrete-event dynamic programming operator to construct the value function and study its properties. A by-product of this approach is the analysis of the differentiability of the value function. We show that differentiability may break down for example with discontinuous arrival intensities. Therefore, one should exercise caution in using arguments based on the differentiability of the value function and the Hamilton–Jacobi–Bellman equation in general.

An exact static solution approach for the service parts end-of-life inventory problem

Abstract This paper studies the spare parts end-of-life inventory problem that happens after the discontinuation of part production. A final ordering quantity is set such that the service process is sustained until all service obligations expire. Also, the price erosion of substitutable or new generation products over time makes it economically justifiable to consider switching to an alternative service policy for repair such as swapping the old product with a new one. This requires the joint optimization of the final order quantity and the time to switch from repair to an alternative service policy. To the best of our knowledge, the problem has not been optimally solved yet either in its static or dynamic formulation. In the current paper, we solve its static version as a bi-level optimization problem. We investigate the convexity of the objective function and give a computationally efficient algorithm to find an exact optimal solution up to any given numerical error level ϵ > 0. We illustrate our approach on some numerical examples and compare our results with earlier works on this problem.

A Static Model in Single Leg Flight Airline Revenue Management

Static models on single leg airline revenue management generally consider booking limits or protection limits as the main decision variables to control reservation requests. In the current paper, we provide an alternative framework in which the decision variables are the closing times of fare classes. In a continuous time model with nonhomogeneous Poisson arrivals, cancellations, and no-shows, we study the problem of finding optimal closing times to maximize the expected net revenue from a given flight. We analyze the value function, point out some easy cases, and bring an easily implementable dynamic programming based solution method. We also illustrate this method on some numerical examples.

A mathematical model for personalized advertisement in virtual reality environments

We consider a personalized advertisement assignment problem faced by the manager of a virtual reality environment. In this online environment, users log in/out, and they spend time in different virtual locations while they are online. Every time a user visits a new virtual location, the site manager can show the ad of an advertiser. At the end of a fixed time horizon, the manager collects revenues from all of the advertisers, and the total revenue depends on the number of ads of different advertisers she displays to different users. In this setup, the objective of the manager is to find an optimal dynamic ad display policy in order to maximize her expected revenue. In the current paper, we formulate this problem as a continuous time stochastic optimization problem in which the actions of users are represented with two-state Markov processes and the manager makes display decisions at the transition times of these processes. To our best knowledge, no formal stochastic model and rigorous analysis has been given for this practical problem. Such a model and its analysis are the major contributions of this paper along with an optimal solution.