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-(mu) integral(t)(0) X (s)ds for some mu >= 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(-mu) integral(t)(0) Y (s)ds for some mu >= 1, where Y (t) follows an extended Cox-Ingersoll-Ross model with stochastic reversion level -X (t)/(2 beta) 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. (c) 2013 Elsevier B.V. All rights reserved.