Norkin B. A successive approximation method for solution of actuarial integral equations.

In the dissertation various generalizations of the classical risk process describing a stochastic evolution of the capital of an insurance company (Kramer - Lundberg model) are considered. In particular risk processes with variable deterministic premiums, with random premiums, with non Poisson flows of premiums and claims, risk processes in a stochastic markovian environment are studied. Integral equations for the probability of nonruin as a function of the initial capital of the company for various generalizations of the classical risk process are deduced. For a risk process in a stochastic markovian environment system of integral equations for a set of nonruin probabilities from various initial states of the process are obtained. A general necessary and sufficient, and also concrete sufficient conditions of the existence and uniqueness of solutions of the considered integral equations and systems of equations are established. A successive approximation method for a numerical or analytical solutionof the considered integral equations of insurance mathematics is theoretically and practically validated, in particular its uniform convergence and rate of such convergence are established. A technique of estimation of the accuracy of approximate solutions of integral equations of insurance mathematics is developed by construction of approximations to the exact solution from above and from below. The suggested method of consecutive approximations was tested on a number of numerical examples, its comparison with Monte Carlo method, with known approximations of solutions was carried out. The developed method of successive approximations for the solution of integral equations of insurance mathematics allows to increase accuracy of actuarial calculations, namely, to calculate within a chosen framework a probability of ruin of an insurance company with any prescribed accuracy, to check applicability and accuracy of various empirical approximate formulas for the probability of ruin, if necessary to improve accuracyof empirical approximations in an iterative way, to estimate accuracy and to correct parameters of Monte Carlo simulation method for calculation of the probability of ruin.