×

Duality for a class of continuous-time reversible Markov models. (English) Zbl 1459.60149

Summary: Using a conditional probability structure we build transition probabilities that drive appealing classes of reversible Markov processes. The mechanism used in such a construction allows to find a dual Markov process. This kind of duality is then used to compute the predictor operator of one process via its dual. In particular, we identify the dual of some non-conjugate models, namely the \(M/M/\infty\) queue model and a simple birth, death and immigration process. Such duals ensure that the computation of the predictor operators can be done via finite sums.

MSC:

60J25 Continuous-time Markov processes on general state spaces
60G35 Signal detection and filtering (aspects of stochastic processes)
60K25 Queueing theory (aspects of probability theory)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Karlin, S., A first course in stochastic processes (2014), New York: Academic Press, New York · Zbl 0177.21102
[2] Liggett, TM., Continuous time Markov processes: an introduction, Vol. 113 (2010), Providence, RI: American Mathematical Society, Providence, RI · Zbl 1205.60002
[3] Ethier, SN; Kurtz, TG., Markov processes: characterization and convergence (2009), New York: John Wiley \(####\) Sons
[4] Dynkin, EB., Markov processes (1965), Berlin (HD): Springer, Berlin (HD) · Zbl 0132.37901
[5] Särkkä, S., Bayesian filtering and smoothing (2013), Cambridge,UK: Cambridge University Press · Zbl 1274.62021
[6] Mena, R.; Walker, S., On a construction of Markov models in continuous time, METRON, 67, 303-323 (2009) · Zbl 1416.60074
[7] Papaspiliopoulos, O.; Ruggiero, M., Optimal filtering and the dual processes, Bernoulli, 20, 1999-2019 (2014) · Zbl 1302.60071
[8] Diaconis, P.; Fill, JA., Strong stationary times via a new form of duality, Ann Probab, 18, 4, 1483-1522 (1990) · Zbl 0723.60083
[9] Möhle, M., The concept of duality and applications to Markov processes arising in neutral population genetics models, Bernoulli, 5, 5, 761-777 (1999) · Zbl 0942.92020
[10] Liggett, TM., Interacting particle systems, Vol. 276 (2012), New York: Springer Science & Business Media, New York
[11] Jansen, S.; Kurt, N., On the notion(s) of duality for Markov processes, Probab Surv, 11, 59-120 (2014) · Zbl 1292.60077
[12] Pitt, MK; Chatfield, C.; Walker, S., Constructing first order stationary autoregressive models via latent processes, Scand J Stat, 29, 657-663 (2002) · Zbl 1035.62086
[13] Leisen, F.; Mena, R.; Palma, F., On a flexible construction of a negative binomial model, Stat Probab Lett, 152, 1-8 (2019) · Zbl 1459.60146
[14] Kelly, FP., Reversibility and stochastic networks (2011), Cambridge, UK: Cambridge University Press · Zbl 1260.60001
[15] Lindley, DV., The theory of queues with a single server (1952), Cambridge, UK: Cambridge University Press · Zbl 0046.35501
[16] Cox, J.; Ingersoll, J.; Ross, S., A theory of the term structure of interest rates, Econometrica, 53, 385-407 (1985) · Zbl 1274.91447
[17] Walker, S.; Hatjispyros, S.; Nicoleris, T., A Fleming-Viot proces and Bayesian nonparametrics, Ann Appl Probab, 17, 67-80 (2007) · Zbl 1131.60045
[18] Papaspiliopoulos, O.; Ruggiero, M.; Spanò, D., Conjugacy properties of time-evolving Dirichlet and gamma random measures, Electron J Stat, 10, 3452-3489 (2016) · Zbl 1353.62092
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.