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 > − 1 2 we prove that t − 1 − γ ∫ 0 t x ( s ) d s converges almost surely as t → ∞ . A similar result is also proved for a two-factor affine model.