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Fractional Cox–Ingersoll–Ross process with non-zero «mean»
Volume 5, Issue 1 (2018), pp. 99–111
Yuliya Mishura   Anton Yurchenko-Tytarenko  

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https://doi.org/10.15559/18-VMSTA97
Pub. online: 5 March 2018      Type: Research Article      Open accessOpen Access

Received
26 September 2017
Revised
29 January 2018
Accepted
30 January 2018
Published
5 March 2018

Abstract

In this paper we define the fractional Cox–Ingersoll–Ross process as $X_{t}:={Y_{t}^{2}}\mathbf{1}_{\{t<\inf \{s>0:Y_{s}=0\}\}}$, where the process $Y=\{Y_{t},t\ge 0\}$ satisfies the SDE of the form $dY_{t}=\frac{1}{2}(\frac{k}{Y_{t}}-aY_{t})dt+\frac{\sigma }{2}d{B_{t}^{H}}$, $\{{B_{t}^{H}},t\ge 0\}$ is a fractional Brownian motion with an arbitrary Hurst parameter $H\in (0,1)$. We prove that $X_{t}$ satisfies the stochastic differential equation of the form $dX_{t}=(k-aX_{t})dt+\sigma \sqrt{X_{t}}\circ d{B_{t}^{H}}$, 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 $X_{t}={Y_{t}^{2}}$. 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\to \infty $.

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Keywords
Fractional Cox–Ingersoll–Ross process stochastic differential equation Stratonovich integral

MSC2010
60G22 60H05 60H10

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