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On Jump-Diffusive Driving Noise Sources

Some Explicit Results and Applications

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Abstract

We study some linear and nonlinear shot noise models where the jumps are drawn from a compound Poisson process with jump sizes following an Erlang-m distribution. We show that the associated Master equation can be written as a spatial mth order partial differential equation without integral term. This differential form is valid for state-dependent Poisson rates and we use it to characterize, via a mean-field approach, the collective dynamics of a large population of pure jump processes interacting via their Poisson rates. We explicitly show that for an appropriate class of interactions, the speed of a tight collective traveling wave behavior can be triggered by the jump size parameter m. As a second application we consider an exceptional class of stochastic differential equations with nonlinear drift, Poisson shot noise and an additional White Gaussian Noise term, for which explicit solutions to the associated Master equation are derived.

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Authors and Affiliations

  1. GALATEA, Ecole Polytechnique Fédérale de Lausanne, 1015, Lausanne, Switzerland

    Max-Olivier Hongler

  2. IEM, Bern University of Applied Sciences, 2501, Biel, Switzerland

    Roger Filliger

Authors
  1. Max-Olivier Hongler
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  2. Roger Filliger
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Correspondence to Max-Olivier Hongler.

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Hongler, MO., Filliger, R. On Jump-Diffusive Driving Noise Sources. Methodol Comput Appl Probab 21, 753–764 (2019). https://doi.org/10.1007/s11009-017-9566-3

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  • DOI: https://doi.org/10.1007/s11009-017-9566-3

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