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Reciprocal Class of Jump Processes

Author

Listed:
  • Giovanni Conforti

    (Institut für Mathematik der Universität Potsdam)

  • Paolo Dai Pra

    (Universitá degli Studi di Padova)

  • Sylvie Rœlly

    (Institut für Mathematik der Universität Potsdam)

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.

Suggested Citation

  • Giovanni Conforti & Paolo Dai Pra & Sylvie Rœlly, 2017. "Reciprocal Class of Jump Processes," Journal of Theoretical Probability, Springer, vol. 30(2), pages 551-580, June.
    • Handle: RePEc:spr:jotpro:v:30:y:2017:i:2:d:10.1007_s10959-015-0655-3
    DOI: 10.1007/s10959-015-0655-3
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    References listed on IDEAS

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    1. Roelly, Sylvie & Thieullen, Michèle, 2005. "Duality formula for the bridges of a Brownian diffusion: Application to gradient drifts," Stochastic Processes and their Applications, Elsevier, vol. 115(10), pages 1677-1700, October.
    2. Lévy, Bernard C., 1997. "Characterization of Multivariate Stationary Gaussian Reciprocal Diffusions, ," Journal of Multivariate Analysis, Elsevier, vol. 62(1), pages 74-99, July.
    3. Chay, S. C., 1972. "On quasi-Markov random fields," Journal of Multivariate Analysis, Elsevier, vol. 2(1), pages 14-76, March.
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    Citations

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    Cited by:

    1. Giovanni Conforti & Tetiana Kosenkova & Sylvie Rœlly, 2019. "Conditioned Point Processes with Application to Lévy Bridges," Journal of Theoretical Probability, Springer, vol. 32(4), pages 2111-2134, December.

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