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

Abstract

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.