Partial reconstruction of the trajectories of a discretely observed branching diffusion with immigration and an application to inference [Elektronische Ressource] / Christian Brandt

Partial reconstructionof the trajectoriesof a discretely observedbranching di usion with immigrationandan application to inferenceDissertationzur Erlangung des Grades\Doktor der Naturwissenschaften\der Johannes Gutenberg-Universitatin Mainzam Fachbereich17 Mathematik und InformatikChristian Brandtgeboren in TrierMainz, 2005SummaryIn this treatise we consider nite systems of branching particles where the par-ticles move independently of each other according to d-dimensional di usions.Particles are killed at a position dependent rate , leaving at their death positiona random number of descendants according to a position dependent reproducti-on law p() on =f1g. In addition particles immigrate at constant rate c (one0immigrant per immigration time). A process ’ with above properties is called abranching di usion with immigration (BDI).In the rst part we present the model in detail and discuss the properties of theBDI under our basic assumptions.In the second part we consider the problem of reconstruction of the trajectoryof ’ from discrete observations. We observe the positions of the particles of ’ atdiscrete times t = i for a small step width > 0; in particular we assume thatiwe have no information about the pedigree of the particles.A natural question arises if we want to apply statistical procedures on the dis-crete observations: How can we nd couples of particle positions which belong tothe same particle?