Long-term returns in stochastic interest rate models: different convergence results

Abstract

In this paper, we focus on different convergence results of the long-term return (1/t)∫t0rudu, where the short interest rate r follows an extension of the Cox–Ingersoll–Ross (1985) model. Using the theory of Bessel processes, we proved the convergence almost everywhere of (1/t)∫t0Xudu, where (Xu)u⩾0 denotes a generalization of a Besselsquare process with drift. We also studied the convergence in law of the long-term return in order to make some approximations. We observed the convergence in law of the sequence of processes (Yn)n⩾1 with (Ynt)t⩾0=[(−2β3 δ¯n)1/2∫nt0(Xu+δu 2β) du]t⩾0 By the Aldous criterion, this sequence converges in law to a Brownian motion. These convergence results have some immediate applications. © 1998 John Wiley & Sons, Ltd.