Publication: Numerical evaluation of renewal equations: applications to risk theory and financial models
Full text at PDC
Advisors (or tutors)
Facultad de Ciencias Económicas y Empresariales. Decanato
The so-called Renewal Theory is a frequently used methodology in applied mathematics. Renewal Theory is mainly focussed on solving a Volterra integral equation of the second kind known as Renewal Integral EquationAn interesting problem arises when choosing the appropriate numerical tool in order to approximate the solution of the former integral. The decision will be based on the degree of knowledge of function F(x) and some properties of <I>(u). Three methods based in classical methodologies (simulation, product integration and inverting Laplace transform) will be presented and applied to the calculation of ultimate ruin probabilities in the classical case of Risk Theory. The first one is an original simulation scheme, based on the importance sampling technique, that leads to tight interval estimations of the solution of the Renewal equation. In the second one, the use of the so-called Product Integration technique will be considered and compared with other techniques based on the Newton-Cotes methodology. The last method considered is the Gaver-Stehfest algorithm of inverting Laplace transformo This last one, under certain conditions, could be considered as a very fast and accurate method.
Bahvalov , N.S. (1959). On approximate calculation of multiple integrals, Vestnik Moscov. Univ. Ser. Mat. Meh. Ast. Fiz. Him., 4, 3-8. Bratley, P.; Fox, B.L. & Schrage, L.E. (1987). A guide to simulation, Springer-Verlag, New York. Bühhnann,H. (1970). Mathematical methods in Risk Theory. Springer Verlag, New York. Burden, R.L. and Faires, J.D.(1985). Numerical Analysis, P.W.S., Boston. Davies,B and Martin,B. (1979). Numerical inversion of the Laplace transform: a survey and comparison of methods. Journal of computational physics,33. De Vylder, F. and Goovaerts, M.J. (1988). Recursive Calculation of Finite-time ruin probabilities, Insurance: Mathematics and Economics, 7, 1-7. Delves, L. M. and Mohamed, J. L. (1985). Computational methods for integral equations. Cambridge, England. Cambridge University Press. Feller, W. (1973). An introduction to probability and its applications. Volume JI. John Willey. Fishman, G.S. (1996). Monte Cario: concepts, algorithms and applications. Springer series in operations research. Springer-Verlag, New York. Gaver, D. P. (1966). Operational Research. 14,444-459. Grandell, J. (1990). Aspects of Risk Theory. Springer-Verlag. New York. Haber, S. (1970). Numerical evaiuation of multiple integrals, SIAM Rev., 12,481-526. Nieden-eiter, R. (1978). Quasi-Monte Carlo methods and pseudorandom numbers, Bull. Amer. Math. Soc., 84, 957-1041. Nieden-eiter, R. (1992). Random number generation and Quasi-Monte Carlo methods, Society for industrial and applied mathematics, Philadelphia, PA. Panjer, R.R. (1981). Recursive evaluation of afamily of compound distributions, ASTIN Bulletin, 12, 22-26. Panjer, R.R. & Willmot, G.E. (1992). Insurance risk Models, Society of Actllaries, Schaumburg. Ramsay, C.M. (1992a). "A Practical Algorithm for Approximating the Probability of Ruin." Transactions of the Society of Actuaries, XLIV, 443-59. Ramsay, C.M. (1992b). "Improving Goovaerts' and De Vylder's Stable Recursive Algorithm. ASTIN Bulletin, 22, 51-59. Ramsay,C.M. and Usábel,M.A. (1997).Calculating Ruin probabilities via Product integration. Próxima aparición ASTIN BULLETIN :Mayo 1997.