Corrected Phase-type Approximations of Heavy-tailed Stochastic Models Using Perturbation Analysis



Numerical evaluation of performance measures in heavy-tailed stochastic models is an important and challenging problem. By using as vehicle the classical Cramer-Lundberg risk model, we construct very accurate approximations of such performance measures that provide small absolute and relative errors. Motivated by statistical analysis, we assume that the claim sizes are a mixture of a phase-type and a heavy-tailed distribution and with the aid of perturbation analysis we derive a series expansion for the performance measure under consideration. Our proposed approximations consist of the first two terms of this series expansion, where the first term is a  phase-type approximation of our measure. We refer to our approximations collectively as corrected phase-type approximations. We show that the corrected phase-type approximations exhibit a nice behavior both in finite and infinite time horizon, and we check their accuracy through numerical  experiments.

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