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Asymptotic expansions at work
Abstracts and Reviews
Donsker-Varadhan theory, and other modern developments. We then apply the large deviation theorems to three models in statist_i.cal mechanics, the Curie-Weiss model, the Curie-Weiss-Potts model, and the Ising model. These models are analyzed by the three respective levels of the Donsker-Varadhan theory: the sample means (level l), the empirical measures (level 2), and the empirical processes and fields (level 3). In the last section a general approach to the large deviation analysis of models in statistical mechanics is formulated. Keywords: Large deviation principle, Curie- Weiss model, Curie- Weiss-Potts model, Ising model. 074017 (Mll) Stochastic calculus in actuarial science. ItS’s revolution - our revelation. NorbergR., University of Copenhagen, Denmark, Astin Colloquium Leuven, 1995. The present paper addresses a class of problems in insurance mathematics that all amount to determining the conditional expected value of some functional of some stochastic process.- It demonstrates by examples how modern stochastic calculus opens for extensions of existing results to models of greater diversity and complexity (hence closer resemblance to various real life methodological situations) and represents also unification. The same set of tools are used to determine seemingly unrelated entities from traditionally segregated subfields of actuarial mathematics, including first and higher order moments of present values in life insurance, total claims distributions in non-life insurance, and the probability of ruin in actuarial risk theory. Keywords: Life insurance, non-&e insurance, risk theory, finance, reserves, probability of ruin, point processes, diffusion processes, martingales, stochastic differential equations. 074018 (Mll) A stochastic~ model for the force of interest. Di Lorenzo E., Sibillo M., Blatter der Deutschen Gesellschaftfir Versicherungsmathematik, Vol. 22, nr. 2, 1995, pp. 255-258 The paper provides characterizations for a stochastic model for the force of interest, generated by an Ornstein-Uhlenbeck process, particularly in presence of barriers. Further characterization for the capitalization process are given. Keywords: Ornstein-Uhlenbeck process, first passage time, stochastic force of interest.
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074019 (Mll) Asymptotic expansions at work. Jensen J.L., Institute of Mathematics, Denmark, Scandinavian Actuarial Journal, Harald Cramer Symposium, nr. I, 1995, pp. 143-152. The author has tried to indicate that the theoretical work of Cramer that was mentioned has led to important applications. The applications are naturally based on many developments since the work of Cramer and still more work needs to be done to implement the methods in standard computer packages. Concerning Edgeworth expansions it has been recommended that a stable way of using these is through improved likelihood ratio statistics. This will be sufficient in most cases. If however the extreme tail is of interest or the number of observations is quite small it is recommendable to try to establish a saddlepoint approximation. Keywords: Normal Approximation, Edgeworth Expansions, Saddlepoint Approximations. 074020 (MU, B81) An extension of Kornya’s method with application to pension funds. Dufresne F., University of Lausanne, Switzerland, Astin Colloquium Leuven, 1995. A simple extension of the method of Kornya is derived. The extended method applies to the convolution of triatomic distributions with nonnegative support while the original method is restricted to diatomic distributions. Then, the new algorithm can be applied to the calculation of the total claims distribution of a pension tind where only death and disability of active members are considered. Keywords: Kornya’s method individual model, aggregate claims distributions. 074021(Mll) Bounds for geometric sums in the presence of heavytailed summands. Kalashnikov V.V., Universite de Mame-la-Vallee, France, Prepublications de I’Equipe d’Analyse et de Mathe’matique Applique’es, nr. 27, 1995, pp. I-30. In this paper we obtain two-sided bounds of the distribution function of geometric sums with summands having “heavy tails”. These bounds are obtained with the help of purely probabilistic arguments and do not require complex analytical arguments. Estimates of ruin probabilities in the presence of large claims serve as illustrations of the derived bounds. Keywords: geometric sum, two-sided bounds, renewal
Donsker-Varadhan theory, and other modern developments. We then apply the large deviation theorems to three models in statist_i.cal mechanics, the Curie-Weiss model, the Curie-Weiss-Potts model, and the Ising model. These models are analyzed by the three respective levels of the Donsker-Varadhan theory: the sample means (level l), the empirical measures (level 2), and the empirical processes and fields (level 3). In the last section a general approach to the large deviation analysis of models in statistical mechanics is formulated. Keywords: Large deviation principle, Curie- Weiss model, Curie- Weiss-Potts model, Ising model. 074017 (Mll) Stochastic calculus in actuarial science. ItS’s revolution - our revelation. NorbergR., University of Copenhagen, Denmark, Astin Colloquium Leuven, 1995. The present paper addresses a class of problems in insurance mathematics that all amount to determining the conditional expected value of some functional of some stochastic process.- It demonstrates by examples how modern stochastic calculus opens for extensions of existing results to models of greater diversity and complexity (hence closer resemblance to various real life methodological situations) and represents also unification. The same set of tools are used to determine seemingly unrelated entities from traditionally segregated subfields of actuarial mathematics, including first and higher order moments of present values in life insurance, total claims distributions in non-life insurance, and the probability of ruin in actuarial risk theory. Keywords: Life insurance, non-&e insurance, risk theory, finance, reserves, probability of ruin, point processes, diffusion processes, martingales, stochastic differential equations. 074018 (Mll) A stochastic~ model for the force of interest. Di Lorenzo E., Sibillo M., Blatter der Deutschen Gesellschaftfir Versicherungsmathematik, Vol. 22, nr. 2, 1995, pp. 255-258 The paper provides characterizations for a stochastic model for the force of interest, generated by an Ornstein-Uhlenbeck process, particularly in presence of barriers. Further characterization for the capitalization process are given. Keywords: Ornstein-Uhlenbeck process, first passage time, stochastic force of interest.
233
074019 (Mll) Asymptotic expansions at work. Jensen J.L., Institute of Mathematics, Denmark, Scandinavian Actuarial Journal, Harald Cramer Symposium, nr. I, 1995, pp. 143-152. The author has tried to indicate that the theoretical work of Cramer that was mentioned has led to important applications. The applications are naturally based on many developments since the work of Cramer and still more work needs to be done to implement the methods in standard computer packages. Concerning Edgeworth expansions it has been recommended that a stable way of using these is through improved likelihood ratio statistics. This will be sufficient in most cases. If however the extreme tail is of interest or the number of observations is quite small it is recommendable to try to establish a saddlepoint approximation. Keywords: Normal Approximation, Edgeworth Expansions, Saddlepoint Approximations. 074020 (MU, B81) An extension of Kornya’s method with application to pension funds. Dufresne F., University of Lausanne, Switzerland, Astin Colloquium Leuven, 1995. A simple extension of the method of Kornya is derived. The extended method applies to the convolution of triatomic distributions with nonnegative support while the original method is restricted to diatomic distributions. Then, the new algorithm can be applied to the calculation of the total claims distribution of a pension tind where only death and disability of active members are considered. Keywords: Kornya’s method individual model, aggregate claims distributions. 074021(Mll) Bounds for geometric sums in the presence of heavytailed summands. Kalashnikov V.V., Universite de Mame-la-Vallee, France, Prepublications de I’Equipe d’Analyse et de Mathe’matique Applique’es, nr. 27, 1995, pp. I-30. In this paper we obtain two-sided bounds of the distribution function of geometric sums with summands having “heavy tails”. These bounds are obtained with the help of purely probabilistic arguments and do not require complex analytical arguments. Estimates of ruin probabilities in the presence of large claims serve as illustrations of the derived bounds. Keywords: geometric sum, two-sided bounds, renewal