Modelling a multitype branching Brownian motion: Filtering of a measure-valued process (Q2502274)

The authors study a multitype branching Brownian motion with position-dependent branching law. The model is constructed as a measure-valued process in the framework of marked frees. The dynamics of a two-type system is described as a solution of a martingale problem. Uniqueness of the martingale problem is discussed. A problem is studied regarding information on the whole population when only the cardinality of a subpopulation is observed (e.g., only mutant individuals in a biological species can be observed). This is investigated as a filtering problem. The Kushner-Stratonovich equation for the filter is derived. Uniqueness in law for the solutions of the Kushner-Stratonovich equation is proved by the filtered martingale problem approach. An explicit representation for the filter is obtained via the Feynman-Kac formula.