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A risk model driven by Lévy processes. (English) Zbl 1051.60051
This paper has an introductory character and discusses the ways of incorporating the pure jump Lévy process in risk theory to account for the discrete nature of claims and asset prices as well as semi-heavy tailed claims. The paper deals with the general risk model \(U(t)= u+ct+Z(t)-S(t), t\geq 0\), with the initial reserve \(u\), constant loaded premium \(c\), where the aggregate claims process \(S(t)\) and the random process \(Z(t)\) (representing fluctuation of the risk premium) evolve by jumps. Thus the authors focus on models where \(X(t)=Z(t)-S(t)\) is a pure jump Lévy process and start with a brief survey of the important features of such processes and their relevance to risk modeling. The subordination construction of the Lévy process as a time changed Brownian motion and the concept of business time are used to generalize the classic diffusion model, while the martingale approach yields tractable expressions for the ruin probabilities. Several explicit models of Lévy processes that can be used to drive the risk models are presented, namely, gamma, normal, inverse Gaussian, Meixner and generalized hyperbolic processes. The simulation schemes for various types of Lévy processes are also discussed.

MSC:
60G51 Processes with independent increments; Lévy processes
62P05 Applications of statistics to actuarial sciences and financial mathematics
91B30 Risk theory, insurance (MSC2010)
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