In this paper we define the fractional Cox-Ingersoll-Ross process as Xt :=is a fractional Brownian motion with an arbitrary Hurst parameter H ∈ (0, 1). We prove that Xt satisfies the stochastic differential equation of the form dXt = (k − aXt)dt + σ √ Xt • dB H t , where the integral with respect to fractional Brownian motion is considered as the pathwise Stratonovich integral. We also show that for k > 0, H > 1/2 the process is strictly positive and never hits zero, so that actually Xt = Y 2 t . Finally, we prove that in the case of H < 1/2 the probability of not hitting zero on any fixed finite interval by the fractional Cox-Ingersoll-Ross process tends to 1 as k → ∞.

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.

In this paper the fractional Cox-Ingersoll-Ross process on R+ for H < 1/2 is defined as a square of a pointwise limit of the processes Yε, satisfying the SDE of the form, as ε ↓ 0. Properties of such limit process are considered. SDE for both the limit process and the fractional Cox-Ingersoll-Ross process are obtained.

No abstract

In this paper, we consider option pricing in a framework of the fractional Heston-type model with [Formula: see text]. As it is impossible to obtain an explicit formula for the expectation [Formula: see text] in this case, where [Formula: see text] is the asset price at maturity time and [Formula: see text] is a payoff function, we provide a discretization schemes [Formula: see text] and [Formula: see text] for volatility and price processes correspondingly and study convergence [Formula: see text] as the mesh of the partition tends to zero. The rate of convergence is calculated. As we allow [Formula: see text] to be non-Lipschitz and/or to have discontinuities of the first kind which can cause errors if [Formula: see text] is replaced by [Formula: see text] under the expectation straightforwardly, we use Malliavin calculus techniques to provide an alternative formula for [Formula: see text] with smooth functional under the expectation.