Reciprocal Class of Jump Processes

  • Processes having the same bridges as a given reference Markov process constitute its reciprocal class. In this paper we study the reciprocal class of compound Poisson processes whose jumps belong to a finite set . We propose a characterization of the reciprocal class as the unique set of probability measures on which a family of time and space transformations induces the same density, expressed in terms of the reciprocal invariants. The geometry of plays a crucial role in the design of the transformations, and we use tools from discrete geometry to obtain an optimal characterization. We deduce explicit conditions for two Markov jump processes to belong to the same class. Finally, we provide a natural interpretation of the invariants as short-time asymptotics for the probability that the reference process makes a cycle around its current state.

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Metadaten
Author details:Giovanni ConfortiORCiDGND, Paolo Dai Pra, Sylvie RoellyGND
DOI:https://doi.org/10.1007/s10959-015-0655-3
ISSN:0894-9840
ISSN:1572-9230
Title of parent work (English):Journal of theoretical probability
Publisher:Springer
Place of publishing:New York
Publication type:Article
Language:English
Date of first publication:2015/11/24
Publication year:2015
Release date:2022/05/09
Tag:Compound Poisson processes; Jump processes; Reciprocal processes; Stochastic bridges
Volume:30
Number of pages:30
First page:551
Last Page:580
Funding institution:Berlin Mathematical School; Research Training Group 1845 Stochastic Analysis with Applications in Biology, Finance and Physics
Organizational units:Mathematisch-Naturwissenschaftliche Fakultät / Institut für Mathematik
DDC classification:5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik
Peer review:Referiert
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