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Multitype branching processes observing particles of a given type

Part of: Stochastic systems and control Markov processes

Published online by Cambridge University Press:  14 July 2016

Claudia Ceci  and
Anna Gerardi
Claudia Ceci*
Affiliation:
Università di Chieti
Anna Gerardi*
Affiliation:
Università dell'Aquila
*
∗Postal address: Dipartimento di Scienze, Facoltà di Economia, Università di Chieti, 65127 Pescara, Italy.
∗∗Postal address: Dipartimento di Ingegneria Elettrica, Facoltà di Ingegneria, Università dell'Aquila, L'Aquila, Italy. Email address: gerardi@ing.univaq.it
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Abstract

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A multitype branching process is presented in the framework of marked trees and its structure is studied by applying the strong branching property. In particular, the Markov property and the expression for the generator are derived for the process whose components are the numbers of particles of each type. The filtering of the whole population, observing the number of particles of a given type, is discussed. Weak uniqueness for the filtering equation and a recursive structure for the linearized filtering equation are proved under a suitable assumption on the reproduction law.

Keywords

Branching processmarked treefiltering

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60J75: Jump processes 93E11: Filtering
Type
Research Papers
Information
Journal of Applied Probability , Volume 42 , Issue 2 , June 2005 , pp. 446 - 462
Copyright
© Applied Probability Trust 2005 

References

Athreya, K. B. and Ney, P. E. (1972). Branching Processes. Springer, New York.CrossRefGoogle Scholar
Brémaud, P. (1981). Point Processes and Queues. Springer, New York.CrossRefGoogle Scholar
Calzolari, A. and Nappo, G. (2001). Filtering of a model with grouped data and counting observations. Preprint. Available at http://www.mat.uniroma1.it/people/nappo/nappo.html.Google Scholar
Ceci, C. and Gerardi, A. (2001). Nonlinear filtering equation of a Jump process with counting observations. Acta Appl. Math. 66, 139–154.CrossRefGoogle Scholar
Ceci, C. and Gerardi, A. (2002). Conditional law of a branching process observing a subpopulation. J. Appl. Prob. 39, 112–122.CrossRefGoogle Scholar
Ceci, C. and Mazliak, L. (1992). Une propriété forte de branchements. C. R. Acad. Sci. Paris Sér. I Math. 315, 851–853.Google Scholar
Ceci, C., Gerardi, A. and Mazliak, L. (1996). Some results about stopping times on the marked tree space. Theory Prob. Appl. 41, 578–590.Google Scholar
Chauvin, B. (1991). Product martingales and stopping lines for branching Brownian motion. Ann. Prob. 19, 1195–1205.CrossRefGoogle Scholar
Elliott, R. J. (1982). Stochastic Calculus and Applications (Appl. Math. 18). Springer, New York.Google Scholar
Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes. Characterization and Convergence. John Wiley, New York.CrossRefGoogle Scholar
Jacod, J. (1975). Multivariate point processes: predictable projection, Radon–Nikodým derivatives, representation of martingales. Z. Wahrscheinlichkeitsth. 31, 235–253.CrossRefGoogle Scholar
Kliemann, W., Koch, G. and Marchetti, F. (1990). On the unnormalized solution of the filtering problem with counting process observations. IEEE Trans. Inf. Theory 36, 1415–1425.CrossRefGoogle Scholar
Kurtz, T. G. and Ocone, D. (1988). Unique characterization of conditional distribution in nonlinear filtering. Ann. Prob. 16, 80–107.CrossRefGoogle Scholar
Neveu, J. (1986). Arbres et processus de Galton–Watson. Ann. Inst. H. Poincaré Prob. Statist. 22, 199–207.Google Scholar