Conditional law of a branching process observing a subpopulation

Ceci, Claudia; Gerardi, A.

doi:10.1017/s0021900200021550

Cited by 3 publications

(11 citation statements)

References 7 publications

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“…In order to perform this computation, we need two preliminary results, which are generalizations of Propositions 3.1 and 3.2 of [5].…”

Section: Dynamics Of the Modelmentioning

confidence: 99%

“…To this end, following a procedure suggested in [3] and already used in [5], we introduce the sequence of point processes {v h t } h≥0 given by…”

Section: The Kushner-stratonovich Equationmentioning

confidence: 99%

“…In [5] a homogeneous population was considered, and the observed subpopulation consisted of the descendants of a given particle. In this paper, we deal with a heterogeneous population and we observe only the cardinality of the subpopulation consisting of type-2 particles.…”

mentioning

confidence: 99%

“…Since Y t is not a counting process, by following a method proposed in [3] and already used in [5], we introduce a multivariate point process whose past history coincides with the past history of the observation process. Thus, we are able to write down the filtering equation and then deal with the problem of establishing its uniqueness.…”

mentioning

confidence: 99%

“…Finally, let us observe that the filtering equation for the model studied in [5] was a linear one. This is not the case for a multitype model.…”

mentioning

confidence: 99%

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Multitype branching processes observing particles of a given type

Ceci

Gerardi

2005

J. Appl. Probab.

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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.

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“…In order to perform this computation, we need two preliminary results, which are generalizations of Propositions 3.1 and 3.2 of [5].…”

Section: Dynamics Of the Modelmentioning

confidence: 99%

“…To this end, following a procedure suggested in [3] and already used in [5], we introduce the sequence of point processes {v h t } h≥0 given by…”

Section: The Kushner-stratonovich Equationmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

“…Finally, let us observe that the filtering equation for the model studied in [5] was a linear one. This is not the case for a multitype model.…”

mentioning

confidence: 99%

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Multitype branching processes observing particles of a given type

Ceci

Gerardi

2005

J. Appl. Probab.

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Modelling a Multitype Branching Brownian Motion: Filtering of a Measure-Valued Process

Ceci

Gerardi

2006

Acta Appl Math

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In the framework of marked trees, a multitype branching brownian motion, described by measure-valued processes, is studied. By applying the strong branching property, the Markov property and the expression of the generator are derived for the process whose components are the measure-valued processes associated to each type particles. The conditional law of the measure-valued process describing the whole population observing the cardinality of the subpopulation of a given type particles is characterized as the unique weak solution of the Kushner< Stratonovich equation. An explicit representation of the filter is obtained by Feyman-Kac formula using the linearized filtering equation.

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