Self‐similar processes in collective risk theory

Abstract

Collective risk theory is concerned with random fluctuations of the total assets and the risk reserve of an insurance company. In this paper we consider self-similar, continuous processes with stationary increments for the renewal model in risk theory. We construct a risk model which shows a mechanism of long range dependence of claims. We approximate the risk process by a self similar process with drift. The ruin probability within finite time is estimated for fractional Brownian motion with drift. A similar model is applicable in queueing systems, describing long range dependence in on/off processes and associated fluid models. The obtained results are useful in communication network models, as well as storage and inventory models

Parisian ruin of self-similar Gaussian risk processes

“…Consequently, (28) and (31)(32)(33) Further, in view of (13) and (24) we have Therefore, the claim of Theorem 5.1 follows with F α (λ) ∈ (0, ∞).…”

“…Below we shall refer to R u as the self-similar Gaussian risk process. The justification for choosing self-similar processes to model the aggregated claim process comes from [32], where it is shown that the ruin probability for self-similar Gaussian risk processes is a good approximation of the ruin probability for the classical risk process. Recent contributions have shown that self-similar Gaussian processes such as fractional Brownian motion (fBm), sub-fractional Brownian motion and bi-fractional Brownian motion are useful in modeling of financial risks, see e.g., [18,24,25,28,37] and the references therein.…”

Ruin probability at a given time for a model with liabilities of the fractional Brownian motion type: A partial differential equation approach

“…The case H B/1/2 will correspond to negative correlation between claims. An example of a risk process with long range dependence was developed by Michna [8], who constructed a risk model in which claims appear in good and bad periods (e.g. good weather and bad weather), and under the assumption that the claims in bad periods are bigger than the claims of the good periods.…”

“…Most of these works, to the best of our knowledge, deal with the asymptotic properties of ruin probability, using probabilistic techniques and provide upper and lower bounds for the ruin probability in certain limiting situations. For instance, Michna [8,9] investigates ruin probabilities and first passage times for selfsimilar processes. He proposes self-similar processes as a risk model with claims appearing in good and bad periods.…”

Extremes of randomly scaled Gumbel risks

“…Motivated by [18,19] several contributions have investigated the basic properties of the iterated process Z(t) = X(Y (t)), t ≥ 0, see e.g., [20][21][22] and the references therein. One particular instance is X = B H , and thus by the self-similiarity of fractional Brownian motion we have…”