Reciprocal classes of Markov processes : an approach with duality formulae

• Main Author: Murr, Rüdiger
• Format: Others
• Language: English
• Published: Universität Potsdam 2012
• Subjects: Duality formula; reciprocal class; Levy process; infinite divisibility; counting process; Malliavin calculus; Mathematics
• Online Access: http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-63018; http://opus.kobv.de/ubp/volltexte/2012/6301/

In this work we are concerned with the characterization of certain classes of stochastic processes via duality formulae. First, we introduce a new formulation of a characterization of processes with independent increments, which is based on an integration by parts formula satisfied by infinitely divisible random vectors. Then we focus on the study of the reciprocal classes of Markov processes. These classes contain all stochastic processes having the same bridges, and thus similar dynamics, as a reference Markov process. We start with a resume of some existing results concerning the reciprocal classes of Brownian diffusions as solutions of duality formulae. As a new contribution, we show that the duality formula satisfied by elements of the reciprocal class of a Brownian diffusion has a physical interpretation as a stochastic Newton equation of motion. In the context of pure jump processes we derive the following new results. We will analyze the reciprocal classes of Markov counting processes and characterize them as a group of stochastic processes satisfying a duality formula. This result is applied to time-reversal of counting processes. We are able to extend some of these results to pure jump processes with different jump-sizes, in particular we are able to compare the reciprocal classes of Markov pure jump processes through a functional equation between the jump-intensities.