Sequential tracking of a hidden Markov chain using point process observations

Bayraktar, Erhan; Ludkovski, Michael

doi:10.1016/j.spa.2008.09.003

Cited by 27 publications

(28 citation statements)

References 23 publications

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“…The above also shows thatv andw satisfy the dynamic programming equationsv e −rt {(1 − q t )q t − C1 {a=1} } dt + e −rτ (f (X τ ) − K) | X 0 = x , where the supremum is over all stopping times τ adapted to the filtration of (X t ). Inequalities (46)-(47) and arguments similar to those in [BL09] for a related piecewise-deterministic switching problem imply that the value functions (v, w) are the smallest fixed point of L (in the sense of (48)) bigger than w 0 . We conclude that (v, w) correspond to the smallest solution of (9) which completes the proof.…”

Section: Resultsmentioning

confidence: 89%

Exploration and exhaustibility in dynamic Cournot games

Ludkovski

Sircar

2011

Eur. J. Appl. Math

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We study the stochastic effect of resource exploration in dynamic Cournot models of exhaustible resources, such as oil. We first treat the case of a monopolist who may undertake costly exploration to replenish his diminishing reserves. We then consider a stochastic game between such an exhaustible producer and a "green" producer that has access to an inexhaustible but relatively expensive source, such as solar power. The effort control variable is taken to be either continuous or discrete (switching control). In both settings, we assume that new discoveries occur according to a jump process with intensity given by the exploration effort. This leads to a study of systems of nonlinear first order delay ODEs. We derive asymptotic expansions for the case of a small exploration success rate and present some numerical investigations.

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Section: Resultsmentioning

confidence: 89%

Exploration and exhaustibility in dynamic Cournot games

Ludkovski

Sircar

2011

Eur. J. Appl. Math

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“…This framework is related to our previous work on Bayesian detection problems in the context of Poisson-type observations (Bayraktar and Ludkovski 2009, Ludkovski and Sezer 2012, as well as the author's computational tools (Ludkovski 2009(Ludkovski , 2012. The tractability and flexibility of our approach is demonstrated with several examples in Section 6.…”

Section: New Approachmentioning

confidence: 99%

Bayesian Quickest Detection in Sensor Arrays

Ludkovski¹

2012

Sequential Analysis

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Abstract:We study Bayesian quickest detection problems with sensor arrays. An underlying signal is assumed to gradually propagate through a network of several sensors, triggering a cascade of inter-dependent change-points. The aim of the decision-maker is to centrally fuse all available information to find an optimal detection rule that minimizes Bayes risk. We develop a tractable continuous-time formulation of this problem focusing on the case of sensors collecting point process observations and monitoring the resulting changes in intensity and type of observed events. Our approach uses methods of nonlinear filtering and optimal stopping and lends itself to an efficient numerical scheme that combines particle filtering with Monte Carlo based approach to dynamic programming. The developed models and algorithms are illustrated with plenty of numerical examples.

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“…This investigation was continued by Brekke and Øksendal [10], Duckworth and Zervos [13], Yushkevich and Gordienko [46], and Hamadène and Jeanblanc [21] among others for the continuous time case, and by Yushkevich [44]- [45] for the discrete time case. Other optimal switching and impulse control problems, involving hidden Markov chains in the observable jump processes, were recently studied by Bayraktar and Ludkovski [4]- [5].…”

Section: Introductionmentioning

confidence: 99%

Bayesian Switching Multiple Disorder Problems

Gapeev

2016

Mathematics of OR

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This document is the author's final accepted version of the journal article. There may be differences between this version and the published version. You are advised to consult the publisher's version if you wish to cite from it. Bayesian switching multiple disorder problemsPavel V. Gapeev * To appear in Mathematics of Operations ResearchThe switching multiple disorder problem seeks to determine an ordered infinite sequence of times of alarms which are as close as possible to the unknown times of disorders, or change-points, at which the observable process changes its probability characteristics. We study a Bayesian formulation of this problem for an observable Brownian motion with switching constant drift rates. The method of proof is based on the reduction of the initial problem to an associated optimal switching problem for a three-dimensional diffusion posterior probability process and the analysis of the equivalent coupled parabolic-type free-boundary problem. We derive analytic-form estimates for the Bayesian risk function and the optimal switching boundaries for the components of the the posterior probability process.

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Sequential tracking of a hidden Markov chain using point process observations

Cited by 27 publications

References 23 publications

Exploration and exhaustibility in dynamic Cournot games

Exploration and exhaustibility in dynamic Cournot games

Bayesian Quickest Detection in Sensor Arrays

Bayesian Switching Multiple Disorder Problems

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