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Stationary distributions of a semistochastic process with disturbances at random times and random severity.

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Maria Conceição A. Leite

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We consider a semistochastic continuous-time continuous-state space random process that undergoes downward disturbances with random severity occurring at random times. Between two consecutive disturbances, the evolution is deterministic, given by an autonomous ordinary differential equation. The times of occurrence of the disturbances are distributed according to a general renewal process. At each disturbance, the process gets multiplied by a continuous random variable (“severity”) supported on [0, 1). The inter-disturbance time intervals and the severities are assumed to be independent random variables that also do not depend on the history.

We derive an explicit expression for the conditional density connecting two consecutive post-disturbance levels, and an integral equation for the stationary distribution of the post-disturbance levels. We obtain an explicit expression for the stationary distribution of the random process. Several concrete examples are considered to illustrate the methods for solving the integral equations that occur.


Abstract only. Full-text article is available through licensed access provided by the publisher. Published in Nonlinear Analysis: Real World Applications, 13(2), 497-512. doi: 10.1016/j.nonrwa.2011.02.025. Members of the USF System may access the full-text of the article through the authenticated link provided.