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Rice formula for processes with jumps and applications

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Abstract

We extend Rice Formula to a process which is the sum of two independent processes: a smooth process and a pure jump process with finitely many jumps. Formulas for the mean number of both continuous and discontinuous crossings through a fixed level on a compact time interval are obtained. We present examples in which we compute explicitly the mean number of crossings and compare which kind of crossings dominates for high levels. In one of the examples the leading term of the tail of the distribution function of the maximum of the process over a compact time interval as the level goes to infinity is obtained. We end giving a generalization, to the non-stationary case, of Borovkov-Last’s Rice Formula for Piecewise Deterministic Markov Processes.

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Authors and Affiliations

  1. Departamento de Matemática y Estadística del Litoral, Regional Norte, Universidad de la República, Rivera 1350, 50000, Salto, Uruguay

    Federico Dalmao

  2. Centro de Matemática, Facultad de Ciencias, Universidad de la República, Iguá 4225, 11400, Montevideo, Uruguay

    Ernesto Mordecki

Authors
  1. Federico Dalmao
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  2. Ernesto Mordecki
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Correspondence to Federico Dalmao.

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Dalmao, F., Mordecki, E. Rice formula for processes with jumps and applications. Extremes 18, 15–35 (2015). https://doi.org/10.1007/s10687-014-0200-2

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  • DOI: https://doi.org/10.1007/s10687-014-0200-2

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