## Abstract

The aim of this work is to provide the logical sustainability model for defined contribution pension systems (see [1], [2]) in the discrete framework under stochastic financial rate of the pension system fund and stochastic productivity of the active participants. In addition, the model is developed in the assumption of variable mortality tables.

Under these assumptions, the evolution equations of the fundamental state variables, the pension liability and the fund, are provided. In this very general discrete framework, the necessary and sufficient condition of the pension system sustainability, and all the other basic results of the logical sustainability theory, are proved.

In addition, in this work new results on the efficiency of the rule for the stabilization over time of the level of the unfunded pension liability with respect to wages, level that is defined as β indicator, are also proved.