Mathematical expectation and variance of the final value of certain annuities valued with stochastic financial laws

In this paper we obtain the expression of the random variable final value of an annuity-due, its mathematical expectation and its variance, considering a stochastic accumulation law based on a Wiener Process. Due to the fact that stochastic accumulation factors are dependent when, temporally speaking, the intervals when they are applied overlap, we use a stochastic recurrence equation, in order to determine the variance. Then, from this equation and bearing in mind the expression of the variance of the product of two independent random variables, we get a first order lineal recurrence equation in the variance. This equation gives us the variance of the final value of an annuity-due with g + 1 payments depending on the variance of the final value of the annuity-due with g payments. The coefficients of this equation depend on the parameters of the Wiener Process, on the mathematical expectation of the annuity-due of g payments and on the payment g + 1. When we solve this equation, we get the expression of the variance of the final value of every annuity-due. In this paper, we derive the expressions of the mathematical expectation and variance of the final value in the case of annuities-due being unitary, varying lineally and exponentially. Furthermore, we consider annual, fractioned and continuous payments. We study the effects of different fractioned payments and of variations of the parameters of Wiener Process on the mathematical expectation and variance of the final value.