Long-term Returns in Stochastic Interest Rate Models: Different Convergence Results

In this paper, we focus on different convergence results of the long-term return 1 t rudu 0 t ∫ , where the short interest rate r follows an extension of the Cox, Ingersoll and Ross1 model. Using the theory of Bessel processes, we proved the convergence almost everywhere of 1 t Xudu 0 t ∫ , where Xu ( )u≥0 denotes a generalisation 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 Y n ( ) n≥1 with Yt n ( ) t ≥0 = −2β 3 δn Xu + δ u 2β      0 nt ∫ du       t ≥0 . By Aldous' criterion, this sequence converges in law to a Brownian motion. These convergence results have some immediate applications.