A unified study of bounds and asymptotic estimates for renewal equations and compound distributions with applications to insurance risk analysis.

摘要

This thesis consists of a unified study of bounds and asymptotic estimates for renewal equations and compound distributions and gives applications to aggregate claim distributions, stop-loss premium and ruin probabilities with general claim sizes and especially with heavy-tailed distributions.; Chapter 1 presents the probability models of compound distributions and renewal equations in insurance risk analysis and gives the summary of the results of this thesis.; In Chapter 2, we develop a general method to construct analytical bounds for solutions of renewal equations. Two-sided exponential and linear estimates for the solutions are derived by this method. A generalized Cramer-Lundberg condition is proposed and used to obtain bounds and asymptotic formulae with NWU distributions for the solutions.; Chapter 3 discusses tails of a class of compound distributions introduced by Willmot (1994) and gives uniformly sharper bounds, both with the results obtained in Chapter 2 and renewal theory. The technique of stochastic ordering is employed to get simplified bounds for the tails and to correct the errors of the proofs of some previous results.; In Chapter 4, we derive two-sided estimates for tails of a class of aggregate claim distributions, and especially give upper and lower bounds for compound negative binomial distributions both with adjustment coefficients and with heavy-tailed distributions. For the latter case, Dickson's (1994) condition plays the same role as the Cramer-Lundberg condition.; Chapter 5 is devoted to the aging property of compound geometric distributions and its applications to stop-loss premiums and ruin probabilities. By the aging property, general upper and lower bounds for the stop-loss premium of the class of compound distributions discussed in Chapter 3 are derived, which apply to any claim size distribution. Also, two-sided estimates for the stop-loss premium of negative binomial sums are obtained both under the Cramer-Lundberg condition and under Dickson's condition. General upper and lower bounds for ruin probabilities are also considered in this chapter.; Chapter 6 gives a detailed discussion of the asymptotic estimates of tails of convolutions of compound geometric distributions. Asymptotic estimates for these tails are given under light, medium and heavy-tailed distributions, respectively. Applications of these results are given to the ruin probability in the diffusion risk model. Also, two-sided bounds for the ruin probability are derived by a generalized Dickson condition, which applies to any positive claim size distribution. Finally, we give some examples and consider numerical comparisons of bounds with asymptotic estimates.