Convergence of Disturbed Martingales and a Stochastic Model for Annuity Funds

Abstract

Our aim is a stochastic model for the average income of a given annuity fund. Therefore we consider a special kind of stochastic process {X n : n ∈ IN} satisfying a condition of the form: E(X n+1 |F n ) = α n X n + β n [P] for all n ∈ IN. (0.1) In equation (0.1) X n+1 denotes the income of the annuity fund at time n+ 1 (future) and F n is the information about the money market up to time n (present). We wish to find conditions under which such a stochastic process is P-almost surely convergent and to determine whether or not there is a limiting income from our annuity fund. The question of L 1 -convergence is also considered.