# zbMATH — the first resource for mathematics

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)
Full Text:
##### References:
 [1] Grandell, Springer Series in Statistics, in: Aspects of Risk Theory (1991) · Zbl 0717.62100 [2] Asmussen, Advanced Series on Statistical Science and Applied Probability, in: Ruin Probabilities (2000) · doi:10.1142/2779 [3] Kaas, Modern Actuarial Risk Theory (2001) [4] Bertoin, Cambridge Tracts in Mathematics, in: Lévy Processes (1996) [5] Sato, Lévy Processes and Infinitely Divisible Distributions (1999) · Zbl 0973.60001 [6] Barndorff-Nielsen, Lévy Processes-Theory and Applications (2001) [7] Schoutens, Lévy Processes in Finance: Pricing Financial Derivatives (2003) [8] Barndorff-Nielsen OE Normal inverse Gaussian Distributions and the Modeling of Stock Returns 1995 [9] Barndorff-Nielsen, Processes of normal inverse Gaussian type, Finance and Stochastics 2 pp 41– (1998) · Zbl 0894.90011 · doi:10.1007/s007800050032 [10] Rydberg, The normal inverse Gaussian Lévy process: simulation and approximation, Communications on Statistics and Stochastic Models 13 (4) pp 887– (1997) · Zbl 0899.60036 [11] Eberlein, Hyperbolic Distributions in Finance, Bernouilli 1 pp 281– (1995) · Zbl 0836.62107 [12] Eberlein, Mathematical Finance-Bachelier Congress 2000 (2000) [13] Eberlein, Lévy Processes: Theory and Applications (2001) [14] Eberlein, New insights into smile, mispricing and value at risk: The hyperbolic model, Journal of Business 71 (3) pp 371– (1998) · doi:10.1086/209749 [15] Geman, Pure jump Lévy processes for asset price modelling, Journal of Banking and Finance 26 pp 1296– (2002) · doi:10.1016/S0378-4266(02)00264-9 [16] Carr, The fine structure of asset returns, Journal of Business 75 (2) pp 305– (2002) · doi:10.1086/338705 [17] Schoutens W The Meixner Process in Finance 2001 [18] Sørensen, A Semimartingale approach to some problems in risk theory, ASTIN Bulletin 26 (1) pp 15– (1996) [19] LeBlanc, Lévy Processes in finance: A remedy to the non-stationarity of continuous martingales, Finance and Stochastics 2 pp 399– (1998) · Zbl 0909.90025 · doi:10.1007/s007800050047 [20] Grandell, A class of approximations of ruin probabilities, Scandinavian Actuarial Journal (Supp) pp 37– (1977) · Zbl 0384.60057 [21] Furrer, Stable Lévy motion approximation in collective risk theory, Insurance: Mathematics and Economics 20 pp 97– (1997) [22] Monroe, Processes that can be embedded in Brownian motion, The Annals of Probability 6 pp 42– (1978) · Zbl 0392.60057 [23] Carr P Geman H Madan DH Yor M Stochastic volatility for Lévy processes 2003 [24] Chaubey, On the computation of aggregate claims distributions: some new approximations, Insurance: Mathematics and Economics 23 pp 215– (1998) · Zbl 0920.62131 [25] Madan, The variance gamma process and option pricing, European Finance Review 2 pp 79– (1998) · Zbl 0937.91052 · doi:10.1023/A:1009703431535 [26] Barndorff-Nielsen, Exponentially Decreasing Distributions for the Logarithm of Particle Size, Proceedings of the Royal Society of London A (353) pp 401– (1977) [27] Jørgensen, Lecture Notes in Statistics, in: Statistical Properties of the Generalized Inverse Gaussian Distribution (1982) · Zbl 0486.62022 [28] Devroye, Non-uniform Random Variate Generation (1986) [29] Schoutens, Lévy processes, polynomials and martingales, Communications in Statistics and Stochastic Models 14 (1, 2) pp 335– (1998) · Zbl 0895.60050 [30] Schoutens, Lecture Notes in Statistics, in: Stochastic Processes and Orthogonal Polynomials (2000) · Zbl 0960.60076 [31] Grigelionis, Processes of Meixner type, Lith. Mathematical Journal 39 (1) pp 33– (1999) · Zbl 0959.60034 [32] Grigelionis B 2000 [33] Biane, Probability laws related to the Jacobi theta and Riemann zeta functions, and Brownian excursions, Bulletin of American Mathematical Society 38 pp 435– (2001) · Zbl 1040.11061 · doi:10.1090/S0273-0979-01-00912-0 [34] Pitman J Yor M 2000 [35] Barndorff-Nielsen, Infinite divisibility of the hyperbolic and generalized inverse Gaussian distributions, Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete 38 pp 439– (1977) · Zbl 0403.60026 · doi:10.1007/BF00533162 [36] Cherny, Lecture Notes for the Summer School From Lévy Processes to Semimartingales-Recent Theoretical Developments and Applications to Finance, in: Change of Time and Measure for Lévy Processes (2002) [37] Ané, Order flow, transaction clock, and normality of asset returns, The Journal of Finance 55 (5) pp 2259– (2000) · doi:10.1111/0022-1082.00286 [38] Dufresne, Risk theory for the compound Poisson process that is perturbed by diffusion, Insurance: Mathematics and Economics 10 pp 51– (1991) · Zbl 0723.62065 [39] Dufresne, Risk theory and the gamma process, ASTIN Bulletin 22 pp 177– (1991) [40] Raible S Lévy processes in finance: theory, numerics and empirical facts 2000 · Zbl 0966.60044 [41] Schmidli, Cramér-Lundberg approximations for ruin probabilities of risk processes perturbed by diffusion, Insurance: Mathematics and Economics 16 pp 135– (1995) · Zbl 0837.62087 [42] Jacod, Limit Theorems for Stochastic Processes (1987) [43] Prause K The generalized hyperbolic model: estimation, financial derivatives, and risk measures 1999 [44] Asmussen, Approximations of small jumps of Lévy processes with a view towards simulation, Journal of Applied Probability 38 (2) pp 482– (2001) · Zbl 0989.60047 · doi:10.1239/jap/996986757 [45] Barndorff-Nielsen, Normal modified stable processes, Theory of Probability and Mathematical Statistics (2003) · Zbl 1026.60058 [46] Madan, The VG model for share market returns, Journal of Business 63 pp 511– (1990) · doi:10.1086/296519
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.