Corina Constantinescu

Exact and Asymptotic Results for Insurance Risk Models with Surplus-dependent Premiums

In this paper we develop a symbolic technique to obtain asymptotic expressions for ruin probabilities and discounted penalty functions in renewal insurance risk models when the premium income depends on the present surplus of the insurance portfolio. The analysis is based on boundary problems for linear ordinary differential equations with variable coefficients. The algebraic structure of the Greens operators allows us to develop an intuitive way of tackling the asymptotic behavior of the solutions, leading to exponential-type expansions and Cramer-type asymptotics. Furthermore, we obtain closed-form solutions for more specific cases of premium functions in the compound Poisson risk model.

Bonus–Malus systems with Weibull distributed claim severities

Abstract One of the pricing strategies for Bonus–Malus (BM) systems relies on the decomposition of the claims’ randomness into one part accounting for claims’ frequency and the other part for claims’ severity. By mixing an exponential with a Lévy distribution, we focus on modelling the claim severity component as a Weibull distribution. For a Negative Binomial number of claims, we employ the Bayesian approach to derive the BM premiums for Weibull severities. We then conclude by comparing our explicit formulas and numerical results with those for Pareto severities that were introduced by Frangos & Vrontos.

Ruin probabilities in models with a Markov chain dependence structure

In this paper we derive explicit expressions for the probability of ruin in a renewal risk model with dependence among the increments (Z k ) k>0. We study the case where the dependence structure among (Z k ) k>0 is driven by a Markov chain with a transition kernel that can be described via ordinary differential equations with constant coefficients.

Ruin probabilities in classical risk models with gamma claims

Abstract In this paper, we provide three equivalent expressions for ruin probabilities in a Cramér–Lundberg model with gamma distributed claims. The results are solutions of integro-differential equations, derived by means of (inverse) Laplace transforms. All the three formulas have infinite series forms, two involving Mittag–Leffler functions and the third one involving moments of the claims distribution. This last result applies to any other claim size distributions that exhibits finite moments.

Bonus-Malus Systems with Hybrid Claim Severity Distributions

One of the pricing strategies for Bonus-Malus (BM) systems relies on the decomposition of the claims’ randomness, namely, one part accounting for claims’ frequency and the other part for claims’ severity. For a Negative Binomial number of claims, we focused on the modelling of claim severities using a hybrid structure. While Weibull distribution is suggested for constructing smaller-sized claims, Pareto severities employed by (1) were applied to model large ones. A generalised function for the BM premium rates is then derived. Results are illustrated by a numerical example.