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On a piece-wise deterministic Markov process model

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  • Borovkov, K.
  • Novikov, A.

Abstract

We study a piece-wise deterministic Markov process having jumps of i.i.d. sizes with a constant intensity and decaying at a constant rate (a special case of a storage process with a general release rule). Necessary and sufficient conditions for the process to be ergodic are found, its stationary distribution is found in explicit form. Further, the Laplace transform of the first crossing time of a fixed barrier by the process is shown to satisfy a Fredholm equation of second kind. Solution to this equation is given by exponentially fast converging Neumann series; convergence rate of the series is estimated. Our results can be applied to an important reliability problem.

Suggested Citation

  • Borovkov, K. & Novikov, A., 2001. "On a piece-wise deterministic Markov process model," Statistics & Probability Letters, Elsevier, vol. 53(4), pages 421-428, July.
    • Handle: RePEc:eee:stapro:v:53:y:2001:i:4:p:421-428
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    References listed on IDEAS

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    1. Xiaogu, Zheng, 1991. "Ergodic theorems for stress release processes," Stochastic Processes and their Applications, Elsevier, vol. 37(2), pages 239-258, April.
    2. J. Michael Harrison & Sidney I. Resnick, 1976. "The Stationary Distribution and First Exit Probabilities of a Storage Process with General Release Rule," Mathematics of Operations Research, INFORMS, vol. 1(4), pages 347-358, November.
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    Cited by:

    1. Borovkov, Konstantin & Novikov, Alexander, 2008. "On exit times of Lévy-driven Ornstein-Uhlenbeck processes," Statistics & Probability Letters, Elsevier, vol. 78(12), pages 1517-1525, September.

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