Some remarks on the links between the VIR and Gompertz diffusion models and the links between the CIR and Rayleigh diffusion models

Abstract

The main purpose of this work is to establish new links between the stochastic Cox-Ingersoll-Ross (CIR) model and the stochastic Rayleigh diffusion process (SRDP) and the links between the Vasicek Interest Rate (VIR) process and the stochastic Gompertz diffusion process (SGDP). These links focus on elementary stochastic calculus and Itô’s calculus. Firstly, we prove that the square root of the CIR model is a SRDP. Secondly, we prove that the square of the SRDP is a CIR model. Thirdly, we prove that the exponential of the VIR model is a SGDP. Finally, we prove that the logarithm of the SGDP is a VIR model. New computations of the probability transition density function (PTDF) and the trend functions of the processes have quite simple formulations.