The perturbed renewal equation and diffusion type approximation for risk processes. (English. Ukrainian original) Zbl 1004.60086

Silvestrov, D. S.

Theory Probab. Math. Stat. 62, 145-156 (2001); translation from Teor. Jmovirn. Mat. Stat. 62, 134-144 (2000).

This paper deals with the model of perturbed classical risk process, namely, with the model of risk process in series scheme which takes into account possible oscillations of interest rate coefficient of some insurance company, intensities of the Poisson process and values of the claims. It means that these characteristic indices depend on a small parameter \(\varepsilon>0\). The risk process \(X_{\varepsilon}(t),\) hence, also depends on this small parameter \(\varepsilon\). In this case the ruin probability also depends on this parameter. The aim of this paper is to obtain a new version of diffusion approximation for the ruin probabilities for the perturbed risk process. The technique of perturbed renewal equations is used under minimal conditions which are weaker than those in the functional limit theorems. These conditions include only condition of compactness of the second moments for the distributions of claims and the balance condition.

Reviewer: A.V.Swishchuk (Kyïv)

MSC:

60K05	Renewal theory
91B30	Risk theory, insurance (MSC2010)
60K30	Applications of queueing theory (congestion, allocation, storage, traffic, etc.)

Keywords:

risk process; diffusion approximation; perturbed renewal equation