We show how, from a single simulation run, to estimate the ruin probabilities and their sensitivities (derivatives) in a classic insurance risk model under various distributions of the number of claims and the claim size. Similar analysis is given for the tail probabilities of the accumulated...

We study the structure of point processes N with the property that the vary in a finite-dimensional space where [theta]t is the shift and the [sigma]-field generated by the counting process up to time t. This class of point processes is strictly larger than Neuts' class of Markovian arrival...

For risk processes with a general stationary input, a representation formula of ladder height distributions is proved which includes some additional information on process behaviour at the ladder epoch. The proof is short and probabilistic, and utilizes time reversal, occupation measures and...

Kella and Whitt (J. Appl. Probab. 29 (1992) 396) introduced a martingale {Mt} for processes of the form Zt=Xt+Yt where {Xt} is a Lévy process and Yt satisfies certain regularity conditions. In particular, this provides a martingale for the case where Yt=Lt where Lt is the local time at zero of...

Let [psi]i(u) be the probability of ruin for a risk process which has initial reserve u and evolves in a finite Markovian environment E with initial state i. Then the arrival intensity is [beta]j and the claim size distribution is Bj when the environment is in state j[set membership, variant]E....

The waiting time distribution is studied for the Markov-modulated M/G/1 queue with both the arrival rate [beta]i and the distribution Bi of the service time of the arriving customer depending on the state i of the environmental process. The analysis is based on ladder heights and occupation...

Consider the American put and Russian option (Ann. Appl. Probab. 3 (1993) 603; Theory Probab. Appl. 39 (1994) 103; Ann. Appl. Probab. 3 (1993) 641) with the stock price modeled as an exponential Lévy process. We find an explicit expression for the price in the dense class of Lévy processes with...

Consider a random walk or Lévy process {St} and let [tau](u) = inf {t[greater-or-equal, slanted]0 : St u}, P(u)(·) = P(· [tau](u) < [infinity]). Assuming that the upwards jumps are heavy-tailed, say subexponential (e.g. Pareto, Weibull or lognormal), the asymptotic form of the P(u)-distribution of the process {St} up to time [tau](u) is described as u --> [infinity]. Essentially, the results confirm the folklore that level crossing occurs as result of one big jump. Particular sharp conclusions are obtained for...</[infinity]).>

We study the tail asymptotics of the r.v. X(T) where {X(t)} is a stochastic process with a linear drift and satisfying some regularity conditions like a central limit theorem and a large deviations principle, and T is an independent r.v. with a subexponential distribution. We find that the tail...

A key result underlying the theory of MCMC is that any [eta]-irreducible Markov chain having a transition density with respect to [eta] and possessing a stationary distribution [pi] is automatically positive Harris recurrent. This paper provides a short self-contained proof of this fact using...