Padé approximants for finite time ruin probabilities. (English) Zbl 1301.91020
Summary: In this article, we investigate Padé approximants of hyper-exponential to base on the first moments and the matrix-exponential representation of Padé approximated function. An explicit formula is given for the Laplace transform in the time to calculate finite time ruin probabilities of classical Cramer-Lundberg model. This formula generalizes the ultimate ruin probabilities formula of S. Asmussen and T. Rolski [Insur. Math. Econ. 10, No. 4, 259–274 (1992; Zbl 0748.62058)]. To illustrate this formula, several numerical examples with different values \(u\) are given.
MSC:
| 91B30 | Risk theory, insurance (MSC2010) |
| 41A21 | Padé approximation |
| 65C99 | Probabilistic methods, stochastic differential equations |
Keywords:
hyper-exponential; Padé approximants; matrix exponential representation; ruin probabilities; classical Cramer-Lundberg processCitations:
Zbl 0748.62058Software:
EMphtReferences:
| [1] | Asmussen, Søren; Nerman, Olle; Olsson, Marita, Fitting phase-type distributions via the EM algorithm, Scand. J. Stat., 419-441 (1996) · Zbl 0898.62104 |
| [2] | Harrise, C. M.; Michal, W. G., Distribution estimation using Laplace transform, INFORMS J. Comput., 10, 4, 448-458 (1998) |
| [3] | Gonnet, P.; Güttel, S.; Trefethen, L. N., Robust Padé Approximation via SVD, Soc. Ind. Appl. Math., 55, 1, 101-117 (2013) · Zbl 1266.41009 |
| [4] | Ibryaeva, O. L.; Adukov, V. M., An algorithm for computing a Padé approximant with minimal degree denominator, J. Comput. Appl. Math., 237, 1, 529-541 (2013) · Zbl 1255.65043 |
| [5] | Hesthaven, J. S.; Kaber, S. M.; Lurati, L., Padé-Legendre interpolants for Gibbs reconstruction, J. Sci. Comput., 4, 337-359 (2006) · Zbl 1102.65011 |
| [6] | Cox, D. R., A use of complex probabilities in the theory of stochastic processes, Math. Proc. Cambridge Philos. Soc., 51, 313-319 (1955) · Zbl 0066.37703 |
| [7] | Asmussen, S.; O’Cinneide, C. A., Matrix-exponential distribution, (Encyclopedia of Statistical Science Update, vol. 2 (1998)), 435-440 |
| [8] | Botta, R. B.; Harris, C. M.; Marchal, W. G., Characterization of generalized hyperexponential distribution functions, Stoch. Models, 3, 115-148 (2005) · Zbl 0616.62016 |
| [9] | Asmussen, S.; Bladt, M., Renewal theory and queueing algorithms for matrix exponential distributions, (Chakravarthy, S. R.; Alfa, A. S., Matrix-Analytic Methods in Stochastic Models (1997), Marcel Dekker: Marcel Dekker New York), 313-341 · Zbl 0872.60064 |
| [10] | Fackrell, M. V., Characterization of matrix-exponential distribution. Technical Report (2003), School of Applied Mthematics, The University of Adelaide, (Ph.D thesis) |
| [11] | Fackrell, M. V., Fitting with matrix-exponential distributions, Stoch. Models, 21, 377-400 (2005) · Zbl 1065.62097 |
| [12] | Asmussen, Søren; Rolski, Tomasz, Computational methods in risk theory: A matrix-algorithmic approach, Insurance Math. Econom., 259-274 (1991) · Zbl 0748.62058 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
