Negative claim amounts, bessel functions, linear programming and Miller's algorithm

Abstract

This work is concerned with the evaluation of aggregate claims distributions allowing negative claim amounts, as one encounters them in life insurance for example. The main results apply to the collective as well as to the individual model of risk theory. Using the method proposed by Chan (1984), it is shown that some compound distributions with a two-sided claim size distribution satisfy a linear difference equation of infinite order. In the particular compound Poisson case an interesting formula involving generalized modified Bessel functions is derived. Independent differences of two random variables taking values in the non-negative integers are shown to be pseudo compound Poisson in the sense of Hürlimann (1989, 1990). They satisfy also linear difference equations of infinite order. Two approximations are proposed for the evaluation of the distribution function associated to independent differences. The first one extends results obtained by Hürlimann (1985) and Sundt (1986). In particular computation and approximation errors can be controlled using a ‘stop-loss error function’. The second approximation uses a linear program to solve linear difference equations. In the simplest case Miller's algorithm to compute modified Bessel functions is recovered as a special case.