Let Z(t) be a stationary centered Gaussian process with a Markovian structure. In some fluid models, the stationary buffer content V can be expressed as sup t$0 ð R t 0 ZðsÞ ds 2 ctÞ and PðV . uÞ ¼ Ce 2gu ð1 þ oð1ÞÞ: The asymptotic constant C can be expressed by the so called generalized Pickands constants H . In most cases no formula or approximation for C are known. In this paper we show a method of simulation of C by the use of change of measure technique. The method is applicable when Z(t) is a stationary Ornstein-Uhlenbeck process or ZðtÞ ¼ P n j¼1 X j ðtÞ; where ðX 1 ðtÞ; . . .; X n ðtÞÞ is a Gauss-Markov process. Two examples of simulations are included. Moreover we give a formula for a lower bound for generalized Pickands constants.

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The ruin probability in the classical Brownian risk model can be explicitly calculated for both finite and infinite time horizon. This is not the case for the simultaneous ruin probability in the two-dimensional Brownian risk model. Relying on asymptotic theory, we derive in this contribution approximations for both simultaneous ruin probability and simultaneous ruin time for the two-dimensional Brownian risk model when the initial capital increases to infinity.

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

We investigate the tail asymptotic behavior of the sojourn time for a large class of centered Gaussian processes X, in both continuous-and discrete-time framework. All results obtained here are new for the discrete-time case. In the continuous-time case, we complement the investigations of [1, 2] for non-stationary X. A by-product of our investigation is a new representation of Pickands constant which is important for Monte-Carlo simulations and yields a sharp lower bound for Pickands constant.