A polynomial expansion to approximate the ultimate ruin probability in the compound Poisson ruin model

Pierre-Olivier Goffard; Stéphane Loisel; Denys Pommeret

doi:10.1016/j.cam.2015.06.003

A polynomial expansion to approximate the ultimate ruin probability in the compound Poisson ruin model

Pierre-Olivier Goffard ^{1, 2, *} Stéphane Loisel ³ Denys Pommeret ¹

* Corresponding author

1 IML - Institut de mathématiques de Luminy

2 AXA France

3 SAF - Laboratoire de Sciences Actuarielle et Financière

Abstract : A numerical method to approximate ruin probabilities is proposed within the frame of a compound Poisson ruin model. The defective density function associated to the ruin probability is projected in an orthogonal polynomial system. These polynomials are orthogonal with respect to a probability measure that belongs to Natural Exponential Family with Quadratic Variance Function (NEF-QVF). The method is convenient in at least four ways. Firstly, it leads to a simple analytical expression of the ultimate ruin probability. Secondly, the implementation does not require strong computer skills. Thirdly, our approximation method does not necessitate any preliminary discretisation step of the claim sizes distribution. Finally, the coefficients of our formula do not depend on initial reserves.

Keywords : natural exponential families with quadratic variance functions ultimate ruin probability Compound Poisson model orthogonal polynomials Gamma series expansion Laplace transform inversion

Document type :

Journal articles

Domain :

Humanities and Social Sciences / Economies and finances

Humanities and Social Sciences / Business administration

Mathematics [math] / Probability [math.PR]

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A polynomial expansion to approximate the ultimate ruin probability in the compound Poisson ruin model

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A polynomial expansion to approximate the ultimate ruin probability in the compound Poisson ruin model

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