Long time behaviour of stochastic interest rate models

In this paper, we study the long time behaviour of two classes of stochastic interest rate models. Suppose that x(t) is a one-factor interest rate model with positive jumps. For a suitable constant γ>−12 we prove that t−1−γ∫0tx(s)ds converges almost surely as t→∞. A similar result is also proved for a two-factor affine model.