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Marked point processes on the real line

by Günter Last, Günter Last, Andreas Brandt

This book gives a self-contained introduction to the dynamic martingale approach to marked point processes (MPPs). Based on the notion of a compensator, this approach gives a versatile tool for analyzing and describing the stochastic properties of an MPP. In particular, the authors discuss the relationship of an MPP to its compensator and particular classes of MPPs are studied in great detail. The theory is applied to study properties of dependent marking and thinning, to prove results on absolute continuity of point process distributions, to establish sufficient conditions for stochastic ordering between point and jump processes, and to solve the filtering problem for certain classes of MPPs. Although readers are assumed to be familiar with the basic notions of measure, integration, and probability theory, an appendix contains extensive surveys of the theory of conditional distributions and Lebesgue-Stieltjes calculus. Consequently researchers and graduate students in probability will find this an ideal introduction to this topic.