Long-term behavior of stochastic interest rate models with jumps and memory

The long-term interest rates, for example, determine when homeowners refinance their mortgages in mortgage pricing, play a dominant role in life insurance, decide when one should exchange a long bond to a short bond in pricing an option. In this paper, for a one-factor model, we reveal that the long-term return t−μ∫0tX(s)ds for some μ≥1, in which X(t) follows an extension of the Cox–Ingersoll–Ross model with jumps and memory, converges almost surely to a reversion level which is random itself. Such a convergence can be applied in the determination of models of participation in the benefit or of saving products with a guaranteed minimum return. As an immediate application of the result obtained for the one-factor model, for a class of two-factor model, we also investigate the almost sure convergence of the long-term return t−μ∫0tY(s)ds for some μ≥1, where Y(t) follows an extended Cox–Ingersoll–Ross model with stochastic reversion level −X(t)/(2β) in which X(t) follows an extension of the square root process. This result can be applied to, e.g., how the percentage of interest should be determined when insurance companies promise a certain fixed percentage of interest on their insurance products such as bonds, life-insurance and so on.