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Expansions for Markov-modulated systems and approximations of ruin probability. (English) Zbl 0856.60052

A Cox point process is considered. Its random intensity function is a stationary Markov process with a finite state space. Moreover, this point process is allowed to be marked in such a way that given intensity (stepped) function its marks are independent real values with distributions depending on this intensity function. Let \(N\) be such a marked Markov modulated Cox process. The authors are interested in functionals \(\psi(N)\) of the process when its intensity \(\beta^*\) tends to zero. They give formulas for the first two coefficients of the expansion \[ E\psi(N)= a_1 \beta^*+ a_2 (\beta^*)^2+\cdots+ a_n(\beta^*)^n+ o(\beta^*)^n \] in terms of factorial moments of the process in the case when this functional is a ruin functional for a risk process corresponding to the process \(N\). In this case, these factorial moments depend on intensity matrix of the above Markov process, conditional intensities, and jump distributions, and premium rate function.

MSC:

60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
91B30 Risk theory, insurance (MSC2010)
60K25 Queueing theory (aspects of probability theory)
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