TY  - INPR
A1  - Murr, Rüdiger
T1  - Reciprocal classes of Markov processes : an approach with duality formulae
N2  - 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.
T3  - Preprints des Instituts für Mathematik der Universität Potsdam - 1(2012)26 
KW  - Duality formula
KW  - reciprocal class
KW  - Levy process
KW  - infinite divisibility
KW  - counting process
KW  - Malliavin calculus
Y1  - 2012
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-63018
ER  - 
TY  - INPR
A1  - Conforti, Giovanni
A1  - Roelly, Sylvie
T1  - Reciprocal class of random walks on an Abelian group
N2  - Processes having the same bridges as a given reference Markov process constitute its reciprocal class. In this paper we study the reciprocal class of a continuous time random walk with values in a countable Abelian group, we compute explicitly its reciprocal characteristics and we present an integral characterization of it. Our main tool is a new iterated version of the celebrated Mecke's formula from the point process theory, which allows us to study, as transformation on the path space, the addition of random loops. Thanks to the lattice structure of the set of loops, we even obtain a sharp characterization. At the end, we discuss several examples to illustrate the richness of reciprocal classes. We observe how their structure depends on the algebraic properties of the underlying group.
T3  - Preprints des Instituts für Mathematik der Universität Potsdam - 4 (2015) 1 
KW  - reciprocal class
KW  - stochastic bridge
KW  - random walk on Abelian group
Y1  - 2015
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-72604
SN  - 2193-6943
VL  - 4
IS  - 1
PB  - Universitätsverlag Potsdam
CY  - Potsdam
ER  - 
TY  - INPR
A1  - Conforti, Giovanni
A1  - Léonard, Christian
A1  - Murr, Rüdiger
A1  - Roelly, Sylvie
T1  - Bridges of Markov counting processes : reciprocal classes and duality formulas
N2  - Processes having the same bridges are said to belong to the same reciprocal class. In this article we analyze reciprocal classes of Markov counting processes by identifying their reciprocal invariants and we characterize them as the set of counting processes satisfying some duality formula.
T3  - Preprints des Instituts für Mathematik der Universität Potsdam - 3 (2014) 9 
KW  - counting process
KW  - bridge
KW  - reciprocal class
KW  - duality formula
Y1  - 2014
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-71855
SN  - 2193-6943
VL  - 3
IS  - 9
PB  - Universitätsverlag Potsdam
CY  - Potsdam
ER  -