We used the random walk to model the problem of reserves. The classic case of a stochastic process is the example of random walks, which are used to study a set of phenomena and, particularly, as in this article, to study models of reserves evolution. Random walks also allow the construction of significant complex systems and are used as an instrument of analysis, being used in this study for giving a theoretical characteristic to specific types of systems. Our goal is mainly to study reserves to see how to ensure that pension funds are sustainable. This paper, by covering a classic approach to the study of pension funds, makes possible to draw interesting conclusions about the problem of reserves. We also consider the Brownian motion to model the pensions fund assets and liability management politics. In this context, it was possible to obtain expressions for the expected value of the pensions fund perpetual maintenance cost present value, also for the expected value of the maintenance cost up to time t, indicators of a fund maintenance policy expenditures.

This work is financed by national funds through FCT - Fundação para a Ciência e Tecnologia, I.P., under the project FCT UIDB/04466/2021. Furthermore, the author(s) thank the Iscte-Instituto Universitário de Lisboa and ISTAR-IUL, for their support.

Brownian motions,Gambler’s ruin,Pensions fund,random walks,reserves

• Mathematics - Natural Sciences
• Computer and Information Sciences - Natural Sciences