A class of very regular distribution functions and corresponding ruin probabilities

Abstract

Abstract In this note, a function is said to be very regular it it has a power series expansion ƒ(x) = ∑ a n x n valid in the whole complex plane and if, moreover, the series ∑ n! a n x n has a strictly positive radius of convergence. In fact, the domain of the considered functions ƒ(x) is restricted to [0, ∞). The class of very regular functions is rather narrow, but the fact that any linear combination of functions xne−az (n: nonnegative integer) is a very regular function, already justifies its consideration. Let S(x) be the distribution function of a single claim in the classical risk model. Let ,,(x) be the ruin probability corresponding to the initial risk reserve x. We prove that, if S(x) is very regular. then ψ(x) is also very regular. Moreover. then the coefficients bn in ψ(x) = ∑ bnx n can be calculated iteratively from the coefficients an in 1 −S(x) = ∑ anx n. Since very regular functions have quickly convergent power series expansions. we possess an easy method for the numerical computatio...