Marginal-sum equations and related fixed-point problems

A marginal-sum equation of order p≥2p≥2 is a system of nonlinear equations which in turn are linear equations for polynomials of degree pp in pp variables. Marginal-sum equations typically arise in the construction of a multiplicative tariff in actuarial mathematics.In the present paper we study the existence and the radial uniqueness of solutions of marginal-sum equations and the possibility of computing solutions by iteration. To this end, we first show that the marginal-sum problem is equivalent to several fixed-point problems and we then study these fixed-point problems and the corresponding fixed-point iterations.We show, as a general result, that a marginal-sum equation always has a solution and that the solution cannot be unique. Moreover, for the case p=2p=2 we show that the solution is radially unique and can be computed by fixed-point iteration with respect to a related fixed-point problem and arbitrary initial values. By contrast, for the case p≥3p≥3 we present a numerical example in which for certain initial values the fixed-point iteration is cyclic and hence divergent.