Symbolic calculation of the moments of the time of ruin

Lin and Willmot [Insurance: Mathematics and Economics 27 (2000) 19] derived a set of defective renewal difference equations involving the moments of the time of ruin in the classical insurance risk model. In principal, this allows for evaluation of these moments. Even with their result, however, it becomes apparent that obtaining even the first two moments is complicated due to the recursive nature of the relation and sheer volume of algebra. To combat these difficulties and obtain higher order moments, we explore the use of symbolic computation as a tool to automate the algebraic detail. In particular, we present a Mathematica program that algebraically calculates the moments of the time of ruin.