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Theory of Probability and Mathematical Statistics

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


Authors: Yu. S. Mishura, V. I. Piterbarg, K. V. Ralchenko and A. Yu. Yurchenko-Tytarenko
Translated by: N. N. Semenov
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 97 (2017).
Journal: Theor. Probability and Math. Statist. 97 (2018), 167-182
MSC (2010): Primary 60G22; Secondary 60G15, 60H10
DOI: https://doi.org/10.1090/tpms/1055
Published electronically: February 21, 2019
MathSciNet review: 3746006
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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.


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Additional Information

Yu. S. Mishura
Affiliation: Department of Probability Theory, Statistics, and Actuarial Mathematics, Faculty for Mechanics and Mathematics, Taras Shevchenko National University of Kyiv, Volodymyrs’ka Street, 64/13, Kyiv 01601, Ukraine
Email: myus@univ.kiev.ua

V. I. Piterbarg
Affiliation: Laboratory of Probability Theory, Faculty for Mechanics and Mathematics, Moscow State University, Leninskie gory, 1, Moscow, 119991, Russian Federation
Email: piter@mech.math.msu.su

K. V. Ralchenko
Affiliation: Department of Probability Theory, Statistics, and Actuarial Mathematics, Faculty for Mechanics and Mathematics, Taras Shevchenko National University of Kyiv, Volodymyrs’ka Street, 64/13, Kyiv 01601, Ukraine
Email: k.ralchenko@gmail.com

A. Yu. Yurchenko-Tytarenko
Affiliation: Department of Probability Theory, Statistics, and Actuarial Mathematics, Faculty for Mechanics and Mathematics, Taras Shevchenko National University of Kyiv, Volodymyrs’ka Street, 64/13, Kyiv 01601, Ukraine
Email: ayurty@gmail.com

DOI: https://doi.org/10.1090/tpms/1055
Keywords: Fractional {\KIR} process, stochastic differential equation, fractional {\OU} process, Stratonovich integral
Received by editor(s): April 23, 2017
Published electronically: February 21, 2019
Article copyright: © Copyright 2019 American Mathematical Society