Stochastic representation and pathwise properties of fractional Cox-Ingersoll-Ross process

Abstract

We consider the fractional Cox-Ingersoll-Ross process satisfying the stochastic differential equation (SDE) $dX_t = aX_t\,dt + \sigma \sqrt{X_t}\,dB^H_t$ driven by a fractional Brownian motion (fBm) with Hurst parameter exceeding $\frac{2}{3}$. The integral $\int_0^t\sqrt{X_s}dB^H_s$ is considered as a pathwise integral and is equal to the limit of Riemann-Stieltjes integral sums. It is shown that the fractional Cox-Ingersoll-Ross process is a square of the fractional Ornstein-Uhlenbeck process until the first zero hitting. Based on that, we consider the square of the fractional Ornstein-Uhlenbeck process with an arbitrary Hurst index and prove that until its first zero hitting it satisfies the specified SDE if the integral $\int_0^t\sqrt{X_s}\,dB^H_s$ is defined as a pathwise Stratonovich integral. Therefore, the question about the first zero hitting time of the Cox-Ingersoll-Ross process, which matches the first zero hitting moment of the fractional Ornstein-Uhlenbeck process, is natural. Since the latter is a Gaussian process, it is proved by the estimates for distributions of Gaussian processes that for $a<0$ the probability of hitting zero in finite time is equal to 1, and in case of $a>0$ it is positive but less than 1. The upper bound for this probability is given.