The estimation of P(S-n u) by simulation, where S, is the sum of independent. identically distributed random varibles Y-1,..., Y-n, is of importance in many applications. We propose two simulation estimators based upon the identity P(S-n u) = nP(S, u, M-n = Y-n), where M-n = max(Y-1,...,...

Well-behaved densities are typically log-convex with heavy tails and log-concave with light ones. We discuss a benchmark for distinguishing between the two cases, based on the observation that large values of a sum X1 + X2 occur as result of a single big jump with heavy tails whereas X1,X2 are...

Phase-type (PH) distributions are defined as distributions of lifetimes of finite continuous-time Markov processes. Their traditional applications are in queueing, insurance risk, and reliability, but more recently, also in finance and, though to a lesser extent, to life and health insurance....

Two insurance companies I 1 ,I 2 with reserves R 1 (t),R 2 (t) compete for customers, such that in a suitable differential game the smaller company I 2 with R 2 (0)<R 1 (0) aims at minimizing R 1 (t)&#x2212;R 2 (t) by using the premium p 2 as control and the larger I 1 at maximizing by using p 1. Deductibles K 1 ,K 2 are fixed but may be different. If K 1 >K 2 and I 2 is the leader choosing its premium first, conditions for Stackelberg equilibrium are established. For gamma-distributed...</r>

In a bonus-malus system in car insurance, the bonus class of a customer is updated from one year to the next as a function of the current class and the number of claims in the year (assumed Poisson). Thus the sequence of classes of a customer in consecutive years forms a Markov chain, and most...

A risk process with constant premium rate $c$ and Poisson arrivals of claims is considered. A threshold $r$ is defined for claim interarrival times, such that if $k$ consecutive interarrival times are larger than $r$, then the next claim has distribution $G$. Otherwise, the claim size...

Algorithms for simulation of a Lévy process X(t) are discussed, with particular emphasis on two algorithms approximating jumps that are in some sense small. One is classical, defining small jumps as those of small absolute value. The other one appears to be new and relies on an completely...

Well-behaved densities are typically log-convex with heavy tails and log-concave with light ones. We discuss a benchmark for distinguishing between the two cases, based on the observation that large values of a sum X1+X2 occur as result of a single big jump with heavy tails whereas X1,X2 are of...

Phase-type (PH) distributions are defined as distributions of lifetimes of finite continuous-time Markov processes. Their traditional applications are in queueing, insurance risk, and reliability, but more recently, also in finance and, though to a lesser extent, to life and health insurance....

Two insurance companies I 1 ,I 2 with reserves R 1 (t),R 2 (t) compete for customers, such that in a suitable differential game the smaller company I 2 with R 2 (0)<R 1 (0) aims at minimizing R 1 (t)−R 2 (t) by using the premium p 2 as control and the larger I 1 at maximizing by using p 1. Deductibles K 1 ,K 2 are fixed but may be different. If K 1 >K 2 and I 2 is the leader choosing its premium first, conditions for Stackelberg equilibrium are established. For gamma-distributed...</r>