Tomasz Rolski

Computational methods in risk theory: A matrix-algorithmic approach

Abstract This paper is concerned with the numerical computation of the probability ψ( u ) of ruin with initial reserve u . The basic assumption states that the claim size distribution is phase-type in the sense of Neuts. The models considered are: the classical compound Poisson risk process, the Sparre Anderse process and varying environments which are either governed by a Markov process or exhibit periodic fluctuations. The computational steps involve the iterative solution of a non-linear matrix equation Q = Ψ ( Q ) as well as the evaluation of matrix-exponentials e Qu . A number of worked-out numerical examples are presented.

Upper Bounds for Single Server Queues with Doubly Stochastic Poisson Arrivals

Consider the following type of single server queues. Arrivals are according to a Doubly Stochastic Poisson process with a stationary, ergodic random intensity {λt}. Service times are independent, identically distributed, also independent from arrivals. It is proven that the mean stationary work-load is not greater than E[ωλ0], where ωa denotes the mean stationary work-load in the M/GI/1 queue with arrival intensity a and the same service process. Similar results are given for the mean stationary queue size and the mean stationary delay.

Queues with nonstationary inputs

In this paper we study single server queues with independent and identically distributed service times and a general nonstationary input stream. We discuss several notions of “being in equilibrium”. For queues with a doubly stochastic Poisson input we survey continuity and bounds of moments of some performance characteristics. We also discuss conjectures posed by Ross [34] to the effect that for a “more stationary” input we have a better performance characteristics. Some results are reviewed to typify a problem and then it is followed by a discussion, questions and related bibliography.

Risk theory in a periodic environment: the Crame´r-Lundberg approximation and Lundberg's inequality

A risk process with the claim arrival intensity β( t ), the claim size distribution β ( t ) and the premium rate p ( t ) at time t being periodic functions of t is considered. It is shown that the adjustment coefficient γ* is the same as for the standard time-homogeneous compound Poisson risk process obtained by averaging the parameters over a period, and a suitable version of the Cramer-Lundberg approximation for the ruin probability ψ ( s ) ( u ) with initial reserve u and initial season s is derived. An approximation in terms of a Markovian environment model with n states is studied, and limit theorems describing the rate of convergence γ n → γ* are given. Finally, various upper and lower bounds of Lundberg type for the ruin probabilities are derived for both the periodic and the Markov-modulated model. By time-reversion, the results apply also to periodic M / G /1 queues.

Simulation of the asymptotic constant in some fluid models

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 and P(V>u)=Ce −γu (1+o(1)). 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 where (X 1(t),…,X n (t)) is a Gauss-Markov process. Two examples of simulations are included. Moreover we give a formula for a lower bound for generalized Pickands constants.

Does Markov-Modulation Increase the Risk?

In this paper we compare ruin functions for two risk processes with respect to stochastic ordering, stop-loss ordering and ordering of adjustment coefficients. The risk processes are as follows: in the Markov-modulated environment and the associated averaged compound Poisson model. In the latter case the arrival rate is obtained by averaging over time the arrival rate in the Markov modulated model and the distribution of the claim size is obtained by averaging the ones over consecutive claim sizes.

Stochastic ordering and thinning of point processes

We study the stochastic ordering of random measures and point processes generated by a partial order [mu]

A Note on Transient Gaussian Fluid Models

In this paper we study the distribution of the supremum over interval [0,T] of a centered Gaussian process with stationary increments with a general negative drift function. This problem is related to the distribution of the buffer content in a transient Gaussian fluid queue Q(T) at time T, provided that at time 0 the buffer is empty. The general theory is illustrated by detailed considerations of different cases for the integrated Gaussian process and the fractional Brownian motion. We give asymptotic results for P(Q(T)>x) and P(sup 0≤t≤TQ(t)>x) as x→∞.

APPROXIMATION OF PERIODIC QUEUES

In this paper we demonstrate how some characteristics of queues with the periodic Poisson arrivals can be approximated by the respective characteristics in queues with Markov modulated input. These Markov modulated queues were recently studied by Regterschot and de Smit (1984). The approximation theorems are given in terms of the weak convergence of some characteristics and their uniform integrability. The approximations are applicable for the following characteristics: mean workload, mean workload at the time of day, mean delay, mean queue size.

A Gaussian fluid model

A fluid model with infinite buffer is considered. The total net rate is a stationary Gaussian process with mean −c and covariance functionR(t). Let Ψ(x) be the probability that in steady state conditions the buffer content exceedsx. Under the condition ∫0∞t2 ¦R(t)¦dt<∞ we show that Ψ admits a logarithmic linear upper bound, i.e. Ψ(x)≤Cexp[−γx]+o(exp[−γx]) and find γ and C. Special cases are worked out whenR is as in a Gauss-Markov or AR-Gaussian process.