Stochastic representation and path properties of a fractional Cox–Ingersoll–Ross process

Abstract

We consider the Cox–Ingersoll–Ross process that satisfies the stochastic differential equation $dX_t = aX_t dt+\sigma \sqrt {X_t} dB^H_t$ driven by a fractional Brownian motion $B^H_t$ with the Hurst index exceeding $\frac {2}{3}$, where $\int _0^t\sqrt {X_s} dB^H_s$ is the pathwise integral defined as the limit of the corresponding Riemann–Stieltjes sums. We show that the Cox–Ingersoll–Ross process coincides with the square of the fractional Ornstein–Uhlenbeck process up to the first return to zero. Based on this observation, we consider the square of the fractional Ornstein–Uhlenbeck process with an arbitrary Hurst index and prove that it satisfies the above stochastic differential equation up to the first return to zero if $\int _0^t\sqrt {X_s} dB^H_s$ is understood as the pathwise Stratonovich integral. Then a natural question arises about the first visit to zero of the fractional Cox–Ingersoll–Ross process which coincides with the first visit to zero of the fractional Ornstein–Uhlenbeck process. Since the latter process is Gaussian, we use the bounds for the distributions of Gaussian processes to prove that the probability of a visit to zero over a finite time equals 1 if $a<0$. Otherwise this probability is positive. We provide an upper bound for this probability.