A stochastic perpetuity takes the form D∞ = ∑∞n=0 exp(Y1 + · · · + Yn)Bn, where (Yn : n ≥ 0) and (Bn : n ≥ 0) are two independent sequences of independent and identically distributed random variables (RVs). This is an expression for the stationary distribution of the Markov chain defined recursively by Dn+1 = AnDn + Bn, n ≥ 0, whereAn = en ;D∞ then satisfies the stochastic fixed-point equationD...

For a random walk with both downward and upward jumps (increments), the joint distribution of the exit time across a given level and the undershoot or overshoot at crossing is determined through its generating function, when assuming that the distribution of the jump in the direction making the exit possible has a Laplace transform which is a rational function. The expected exit time is also de...

In this paper we investigate the number and maximum severity of the ruin excursion of the insurance portfolio reserve process in the Cramér–Lundberg model with and without tax payments. We also provide a relation of the Cramér–Lundberg risk model with the G/G/∞ queue and use it to derive some explicit ruin probability formulae. Finally, the renewal risk model with tax is considered, and an asym...

T subject of probability was, of course, motivated initially by its applications. However, there have been earnest attempts to convert it into a branch of pure mathematics, a development suggested by the axiomatic basis of the theory and reinforced by a preoccupation with possible measure-theoretic pathologies. Excesses in this direction led to a reaction, marked by the appearance of the journa...

We consider an insurance model, where the underlying point process is a Cox process. Using amartingale approach applied to diffusion processes, finite-timeLundberg inequalities are obtained. By change-of-measure techniques, Cramér–Lundberg approximations are derived.

We use the properties of the Matuszewska indices to show asymptotic inequalities for hazard rates. We discuss the relation between membership in the classes of dominatedly or extended rapidly varying tail distributions and corresponding hazard rate conditions. Convolution closure is established for the class of distributions with extended rapidly varying tails.

We prove large deviation results for Minkowski sums Sn of independent and identically distributed random compact sets where we assume that the summands have a regularly varying distribution and finite expectation. The main focus is on random convex compact sets. The results confirm the heavy-tailed large deviation heuristics: ‘large’ values of the sum are essentially due to the ‘largest’ summan...

In this paper we consider a general Lévy process X reflected at a downward periodic barrier At and a constant upper barrier K , giving a process V t = Xt + Lt − Lt . We find the expression for a loss rate defined by l = ELK1 and identify its asymptotics as K → ∞ when X has light-tailed jumps and EX1 < 0.

We consider a stochastic differential equation (SDE) with piecewise linear drift driven by a spectrally one-sided Lévy process. We show that this SDE has some connections with queueing and storage models, and we use this observation to obtain the invariant distribution.