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Stochastic representation and pathwise properties of fractional Cox-Ingersoll-Ross process.ArXiv:1708.02712v1 9Aug2017статья
- Авторы: Yuliya Mishura, Piterbarg Vladimir I., Kostiantyn Ralchenko, Anton Yurchenko-Tytarenko
- Сборник: ArXiv.org.e-Print
- Серия: [math.PR]
- Том: 1708
- Год издания: 2017
- Издательство: Cornell University Press
- Местоположение издательства: United States
- Первая страница: 1
- Последняя страница: 14
- Аннотация: We consider the fractional Cox-Ingersoll-Ross process satisfying the stochastic differential equation (SDE) dXt=aXtdt+σXt−−√dBHt driven by a fractional Brownian motion (fBm) with Hurst parameter exceeding 23. The integral ∫t0Xs−−−√dBHs 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 corresponding integral 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.
- Добавил в систему: Питербарг Владимир Ильич