Projective method for the equation of risk theory in the arithmetic case (Q2440068)

The author considers a discrete ordinary renewal model for the capital of an insurance company where both claim sizes and inter-arrival times are positive integer-valued independent and identically distributed (i.i.d.) random variables which are mutually independent. The initial capital can be any integer value. In this framework, the problem of non-ruin probability is solved by the Wiener-Hopf method. Using generating functions the author reduces the fundamental equation of risk theory to a special one-sided discrete Wiener-Hopf equation. Based on the solvability theory for this equation, a projective method to the approximation of ruin probabilities is applied. Conditions for waiting times and claims under which the method converges are derived. Approximations for the ruin probabilities of the delayed renewal process and the stationary renewal process are obtained.