Stochastic interest rates with actuarial applications

This paper presents recursive double integral equations to obtain the distribution of the discounted value or accumulated value of deterministic cash flows. The double integrals have to be evaluated numerically at each iteration. Those distributions are useful when studying the investment risk of portfolios of insurance contracts. The methods suggested take advantage of the Markovian property of the Gaussian process used to model the future rates of return. We start with the first cash flow and successively add the other cash flows while keeping track of the latest information about the rate of return in order to update the distribution at each step. Various means and covariances of bivariate normal distributions which are required if one wants to apply the results in practice are given. In the paper, the Ornstein–Uhlenbeck process is chosen to model the rate of return but the results could be extended to a second order differential equation.