Method of successive approximations for solving integral equations of actuarial mathematics
The dissertation considers various generalizations of the classical risk process describing the stochastic evolution of the capital of an insurance company (Cramer Lundberg model), including processes with variable deterministic or random premiums, non Poisson flows of premiums and claims, and a stochastic Markovian environment. Integral equations for the probability of nonruin as a function of initial capital are derived for these generalizations. For processes in a stochastic Markovian environment, systems of integral equations for nonruin probabilities corresponding to different initial states are obtained. General necessary and sufficient, as well as specific sufficient, conditions for the existence and uniqueness of solutions are established. A successive approximation method for numerical and analytical solution of the integral equations is theoretically and practically validated; its uniform convergence and rate of convergence are established. A technique for estimating the accuracy of approximate solutions is developed by constructing upper and lower approximations to the exact solution. The method is tested on numerical examples and compared with the Monte Carlo method and known solution approximations. The developed method increases the accuracy of actuarial calculations: it allows the probability of ruin to be calculated with any prescribed accuracy, empirical approximations to be verified and improved iteratively, and the accuracy and parameters of Monte Carlo simulations to be estimated and corrected.