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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.
Author details: | Giovanni ConfortiORCiDGND, Paolo Dai Pra, Sylvie RoellyGND |
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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 |