Reciprocal Class of Jump Processes

Giovanni Conforti (), Paolo Dai Pra () and Sylvie Rœlly ()
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Giovanni Conforti: Institut für Mathematik der Universität Potsdam
Paolo Dai Pra: Universitá degli Studi di Padova
Sylvie Rœlly: Institut für Mathematik der Universität Potsdam

Journal of Theoretical Probability, 2017, vol. 30, issue 2, 551-580

Abstract: Abstract 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 $$\mathcal {A}\subset \mathbb {R}^{d}$$ A ⊂ R d . 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 $$\mathcal {A}$$ A 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.

Keywords: Reciprocal processes; Stochastic bridges; Jump processes; Compound Poisson processes; 60G55; 60H07; 60J75 (search for similar items in EconPapers)
Date: 2017
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DOI: 10.1007/s10959-015-0655-3

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