We consider a risk model with a counting process whose intensity is a Markovian shot-noise process, to resolve one of the disadvantages of the Cramér–Lundberg model, namely the constant intensity of the Poisson process. Due to this structure, we can apply the theory of piecewise deterministic Markov processes on a multivariate process containing the intensity and the reserve process, which allows us to identify a family of martingales. Eventually, we use change of measure techniques to derive an upper bound for the ruin probability in this model. Exploiting a recurrent structure of the shot-noise process, even the asymptotic behaviour of the ruin probability can be determined.

This paper provides a stochastic model, consistent with Solvency II and the Delegated Regulation, to quantify the capital requirement for demographic risk. In particular, we present a framework that models idiosyncratic and trend risks exploiting a risk theory approach in which results are obtained analytically. We apply the model to non-participating policies and quantify the Solvency Capital Requirement for the aforementioned risks in different time horizons.

We propose a generalized Cramér–Lundberg model of the risk theory of non-life insurance and study its ruin probability. Our model is an extension of that of Dubey (1977) to the case of multiple insureds, where the counting process is a mixed Poisson process and the continuously varying premium rate is determined by a Bayesian rule on the number of claims. We use two proofs to show that, for each fixed value of the safety loading, the ruin probability is the same as that of the classical Cramér–Lundberg model and does not depend on either the distribution of the mixing variable of the driving mixed Poisson process or the number of claim contracts.

The probability of successfully spending twice the same bitcoins is considered. A double-spending attack consists in issuing two transactions transferring the same bitcoins. The first transaction, from the fraudster to a merchant, is included in a block of the public chain. The second transaction, from the fraudster to himself, is recorded in a block that integrates a private chain, exact copy of the public chain up to substituting the fraudster-to-merchant transaction by the fraudster-to-fraudster transaction. The double-spending hack is completed once the private chain reaches the length of the public chain, in which case it replaces it. The growth of both chains are modelled by two independent counting processes. The probability distribution of the time at which the malicious chain catches up with the honest chain, or, equivalently, the time at which the two counting processes meet each other, is studied. The merchant is supposed to await the discovery of a given number of blocks after the one containing the transaction before delivering the goods. This grants a head start to the honest chain in the race against the dishonest chain.

The De Vylder and Goovaerts conjecture is an open problem in risk theory, stating that the finite-time ruin probability in a standard risk model is greater than or equal to the corresponding ruin probability evaluated in an associated model with equalized claim amounts. Equalized means here that the jump sizes of the associated model are equal to the average jump in the initial model between 0 and a terminal time T.In this paper, we consider the diffusion approximations of both the standard risk model and its associated risk model. We prove that the associated model, when conveniently renormalized, converges in distribution to a Gaussian process satisfying a simple SDE. We then compute the probability that this diffusion hits the level 0 before time T and compare it with the same probability for the diffusion approximation for the standard risk model. We conclude that the De Vylder and Goovaerts conjecture holds for the diffusion limits.

Chiu and Yin (2005) found the Laplace transform of the last time a spectrally negative Lévy process, which drifts to ∞, is below some level. The main motivation for the study of this random time stems from risk theory: what is the last time the risk process, modeled by a spectrally negative Lévy process drifting to ∞, is 0? In this paper we extend the result of Chiu and Yin, and we derive the Laplace transform of the last time, before an independent, exponentially distributed time, that a spectrally negative Lévy process (without any further conditions) exceeds (upwards or downwards) or hits a certain level. As an application, we extend a result found in Doney (1991).

Let (Yn, Nn)n≥1 be independent and identically distributed bivariate random variables such that the Nn are positive with finite mean ν and the Yn have a common heavy-tailed distribution F. We consider the process (Zn)n≥1 defined by Zn = Yn - Σn-1, where It is shown that the probability that the maximum M = maxn≥1Zn exceeds x is approximately as x → ∞, where F' := 1 - F. Then we study the integrated tail of the maximum of a random walk with long-tailed increments and negative drift over the interval [0, σ], defined by some stopping time σ, in the case in which the randomly stopped sum is negative. Finally, an application to risk theory is considered.

Dufresne et al. (1991) introduced a general risk model defined as the limit of compound Poisson processes. Such model is either a compound Poisson process itself or a strictly increasing Lévy process. Their construction is based on a non-negative non-increasing function Q that governs the jumps of the process. This function, it turns out, is the tail of the Lévy measure of the process. We discuss an illustration of their model using a generalized Inverse Gaussian (GIG) Lévy process. This increasing Lévy process has the gamma and the inverse Gaussian process as particular cases. Although mathematically more complex, the GIG Lévy process keeps some of the nice properties of the simpler gamma process.

We consider an infinite-capacity second-order fluid queue governed by a continuous-time Markov chain and with linear service rate. The variability of the traffic is modeled by a Brownian motion and a local variance function modulated by the Markov chain and proportional to the fluid level in the queue. The behavior of this second-order fluid-flow model is described by a linear stochastic differential equation, satisfied by the transient queue level. We study the transient level's convergence in distribution under weak assumptions and we obtain an expression for the stationary queue level. For the first-order case, we give a simple expression of all its moments as well as of its Laplace transform. For the second-order model we compute its first two moments.

Let C 1, C 2,…,C m be independent subordinators with finite expectations and denote their sum by C. Consider the classical risk process X(t) = x + ct - C(t). The ruin probability is given by the well-known Pollaczek–Khinchin formula. If ruin occurs, however, it will be caused by a jump of one of the subordinators C i . Formulae for the probability that ruin is caused by C i are derived. These formulae can be extended to perturbed risk processes of the type X(t) = x + ct - C(t) + Z(t), where Z is a Lévy process with mean 0 and no positive jumps.

In this paper, insurance claims X on [0, ∞) with tail distributions which are O(x−δ) for some δ > 1 are considered. Markets are assumed arbitrageable, the insurer setting a premium P > E[X]. Setting a premium as a fixed quantile of the loss distribution presents difficulties; for Pareto distributions with F(x) = 1 – (x + l)–δ ‘ultimately' (as δ ↓ 1) E[X] is larger than any quantile. When δ is near 1, premiums determined by weighting outcomes and a rule analogous to the expected utility principle are highly sensitive to change in δ, which is generally unknown or known only approximately. Under these circumstances, to protect insurers' interests, strategies are needed which provide some ‘premium stability' across a range of δ-values. We introduce a class of pricing functions which are functionally dependent on the governing loss distribution, and which are themselves distribution functions. We demonstrate that they provide a coherent framework for pricing insurance premiums when the loss distribution is fat tailed, and enable some degree of premium stability to be established.

We show how good multivariate Poisson mixtures can be approximated by multivariate Poisson distributions and related finite signed measures. Upper bounds for the total variation distance with applications to risk theory and generalized negative multinomial distributions are given. Furthermore, it turns out that the ideas used in this paper also lead to improvements in the Poisson approximation of generalized multinomial distributions.

We reconsider the problem of producing fair and accurate tariffs based on aggregated insurance data giving numbers of claims and total costs for the claims. Jørgensen and de Souza (Scand Actuarial J., 1994) assumed Poisson arrival of claims and gamma distributed costs for individual claims. Jørgensen and de Souza (1994) directly modelled the risk or expected cost of claims per insured unit, μ say. They observed that the dependence of the likelihood function on μ is as for a linear exponential family, so that modelling similar to that of generalized linear models is possible. In this paper we observe that, when modelling the cost of insurance claims, it is generally necessary to model the dispersion of the costs as well as their mean. In order to model the dispersion we use the framework of double generalized linear models. Modelling the dispersion increases the precision of the estimated tariffs. The use of double generalized linear models also allows us to handle the case where only the total cost of claims and not the number of claims has been recorded.

A method for analysing the risk of taking a too low reserve level by the use of Chain Ladder method is developed. We give an answer to the question of how much safety loading in terms of the Chain Ladder standard error has to be added to the Chain Ladder reserve in order to reach a specified security level in loss reserving. This is an important question in the framework of integrated risk management of an insurance company. Furthermore we investigate the relative bias of Chain Ladder estimators. We use Monte Carlo simulation technique as well as the collective model of risk theory in each cell of run-off table. We analyse deviation between Chain Ladder reserves and Monte Carlo simulated reserves statistically. Our results document dependency on claim number and claim size distribution types and parameters.

In this paper we use Fourier/Laplace transforms to evaluate numerically relevant probabilities in ruin theory as an application to insurance. The transform of a function is split in two: the real and the imaginary parts. We use an inversion formula based on the real part only, to get the original function.

By using an appropriate algorithm to compute integrals and making use of the properties of these transforms we are able to compute numerically important quantities either in classical or non-classical ruin theory. As far as the classical model is concerned the problems considered have been widely studied. In what concerns the non-classical model, in particular models based on more general renewal risk processes, there is still a long way to go. In either case the approach presented is an easy method giving good approximations for reasonable values of the initial surplus.

To show this we compute numerically ruin probabilities in the classical model and in a renewal risk process in which claim inter-arrival times have an Erlang(2) distribution and compare to exact figures where available. We also consider the computation of the probability and severity of ruin in the classical model.

Let ℬs([a,b];μ1,μ2,…,μs-1) be the class of all distribution functions of random variables with support in [a,b] having μ1,μ2,…,μs-1 as their first s-1 moments. In this paper we examine some aspects of the structure of ℬs([a,b];μ1,μ2,…,μs-1) and of the s-convex stochastic extrema in it. Using representation results of moment matrices à la Lindsay (1989a), we provide conditions for the admissibility of moment sequences in ℬs([a,b];μ1,μ2,…,μs-1) in terms of lower bounds on the number of support points of the corresponding distribution functions. We point out two special distributions with a minimal number of support points that are the s-convex extremal distributions. It is shown that the support points of these extrema are the roots of some polynomials, and an efficient method for the complete determination of the distribution functions of these extrema is described. A study of the goodness of fit, of the approximation of an arbitrary element in ℬs([a,b];μ1,μ2,…,μs-1) by one of the stochastic s-convex extrema, is then given. Using standard ideas from linear regression, we derive Tchebycheff-type inequalities which extend previous results of Lindsay (1989b), and we establish some limit theorems involving the moment matrices. Finally, we describe some applications in insurance theory, namely, we provide bounds on Lundberg's coefficient in risk theory, and on the actual interest rate relating to a life insurance policy. These bounds are obtained with the aid of the s-convex extrema, and are determined only by the support and the first few moments of the underlying distribution.

In applied probability, the distribution of a sum of n independent Bernoulli random variables with success probabilities p1,p2,…, pn is often approximated by a Poisson distribution with parameter λ = p1 + p2 + pn. Popular bounds for the approximation error are excellent for small values, but less efficient for moderate values of p1,p2,…,pn.

Upper bounds for the total variation distance are established, improving conventional estimates if the success probabilities are of medium size. The results may be applied directly, e.g. to approximation problems in risk theory.

The paper considers stability and instability properties of the Markov chain generated by the composition of an i.i.d. sequence of random transformations. The transformations are assumed to be either linear mappings or else mappings which can be well approximated near 0 by linear mappings. The main results concern the risk probabilities that the Markov chain enters or exits certain balls centered at 0. An application is given to the probability of extinction in a model from population dynamics.

Let ?(u) be the probability of eventual ruin in the classical Sparre Andersen model of risk theory if the initial risk reserve is u. For a large class of such models ?(u) behaves asymptotically like a multiple of exp (–Ru) where R is the adjustment coefficient; R depends on the premium income rate, the claim size distribution and the distribution of the time between claim arrivals. Estimation of R has been considered by many authors. In the present paper we deal with confidence bounds for R. A variety of methods is used, including jackknife estimation of asymptotic variances and the bootstrap. We show that, under certain assumptions, these procedures result in interval estimates that have asymptotically the correct coverage probabilities. We also give the results of a simulation study that compares the different techniques in some particular cases.

We consider a dam in which the release rate depends both on the state and some modulating process. Conditions for the existence of a limiting distribution are established in terms of an associated risk process. The case where the release rate is a product of the state and the modulating process is given special attention, and in particular explicit formulas are obtained for a finite state space Markov modulation.