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

Abstract

We consider the fractional Cox-Ingersoll-Ross process satisfying the stochastic differential equation (SDE) dXt = aXt dt + σ √ Xt dB H t driven by a fractional Brownian motion (fBm) with Hurst parameter exceeding 2 3 . The integral t 0 √ XsdB 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 t 0 √Xs 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.

Fractional Cox–Ingersoll–Ross process with non-zero «mean»

“…Remark 1. In the case of k = 0, the process (5) is the fractional Ornstein-Uhlenbeck process and the definition coincides with the one given in [16].…”

Fractional Ornstein-Uhlenbeck Process with Stochastic Forcing, and its Applications

“…With the help of these facts, and applying the representation of covariance function from [25], we can write it for any H ∈ (0, 1) and t ≥ s ≥ 0 in the following nonsymmetric w.r.t. s and t form that permits to avoid the absolute values of the time differences:…”

Fractional Cox–Ingersoll–Ross process with small Hurst indices

“…A simpler pathwise approach is presented in [17] and [18]. There, the fractional Cox-Ingersoll-Ross process was defined as the square of the solution of the SDE…”

“…However, such a definition has a significant disadvantage: according to it, the process remains on the zero level after reaching the latter, and if H < 1/2, such case cannot be excluded. In this paper we generalize the approach presented in [18] and [17] in order to solve this issue.…”

“…In Section 1 we give preliminary remarks on some results concerning the existence and uniqueness of solutions of SDEs driven by an additive fractional Brownian motion, as well as explain connection of this paper to [17] and [18].…”