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Theory of Probability and Mathematical Statistics
(Teoriya imovirnostey ta matematychna statystyka)



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

Yu. S. Mishura, V. I. Piterbarg, K. V. Ralchenko, A. Yu. Yurchenko-Tytarenko

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Abstract: We consider the fractional Cox-Ingersoll-Ross process satisfying the stochastic differential equation (SDE) driven by a fractional Brownian motion (fBm) with Hurst parameter exceeding frac2/3. The integral is pathwise and equals a 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 hitting zero. Based on that, we consider the square of the Ornstein-Uhlenbeck process with an arbitrary Hurst index and prove that before hitting zero it satisfies the specified SDE if the integral is defined as a pathwise Stratonovich integral. Therefore the question about the first time of hitting zero of the Cox-Ingersoll-Ross process, which matches the first time of hitting zero of the fractional Ornstein-Uhlenbeck process, is natural. Since the latter is a Gaussian process, it is proved using the estimates for distributions of Gaussian processes that in case of a<0 the probability of hitting zero in finite time equals 1, and in case of a>0 it is positive but less than 1. The upper estimate of this probability is given.

Keywords: Fractional Cox-Ingersoll-Ross process, stochastic differential equation, fractional Ornstein-Uhlenbeck process, Stratonovich integral

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