Diffusion and Brownian motion processes in modeling the costs of supporting non-autonomous pension funds

Abstract

In this chapter, we consider pensions funds not sufficiently auto financed and systematically maintained with an outside financing effort, usually non-autonomous pension’s funds. This financial effort, made by the managing entity, translates as capital injections into the fund. The objective of this work is to develop a tool that allows predicting the appropriate moments to carry out these interventions and the respective amounts. So, we propose to represent the unrestricted reserves value process of this kind of funds, through a time homogeneous diffusion process with finite expected time till the ruin. A financial tool that regenerates the diffusion is also admitted, at some level with positive value every time it hits a barrier at the origin. Then the financing effort may be modeled as a renewal-reward process if the regeneration level is kept constant. The perpetual maintenance cost expected values evaluation and of the finite time maintenance cost are studied. Then, we focus on a particular situation of this approach, arising when the unrestricted reserves value process behaves as a generalized Brownian motion process.