The case of certain pensions funds that are not auto financed, and are systematically maintained with an outside financing effort, is considered in this work. As a representation of the unrestricted reserves value process of this kind of funds, a time homogeneous diffusion process with finite expected time to ruin is proposed. Then it is admitted a financial tool that regenerates the diffusion at some level with positive value, every time the diffusion hits a barrier at the origin. So, the financing effort can be modeled as a renewal-reward process if the regeneration level is kept constant. The evaluation of the perpetual maintenance cost expected values and of the finite time maintenance cost are studied. Also, we present an application of this approach when the unrestricted reserves value process behaves as a generalized Brownian motion process.

This work is financed by national funds through FCT - Fundação para a Ciência e Tecnologia, I.P., under the project FCT UIDB/04466/2020. Furthermore, the author thanks the Iscte-Instituto Universitário de Lisboa and ISTAR-IUL, for their support.

Pensions fund,Diffusion process,First passage times,Perpetuity,Renewal equation

• Mathematics - Natural Sciences
• Economics and Business - Social Sciences