Hypothesis testing of the drift parameter sign for fractional Ornstein–Uhlenbeck process

Kukush, Alexander; Mishura, Yuliya; Ralchenko, Kostiantyn

doi:10.1214/17-ejs1237

Cited by 16 publications

(8 citation statements)

References 18 publications

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“…that is the same value for the covariance as in the right-hand side of (21). Concerning (iii), let us work with t ≥ s ≥ 0, since R H (t, s) is symmetric in t and s. We have to distinguish three cases.…”

Section: 42mentioning

confidence: 99%

Fractional Ornstein-Uhlenbeck Process with Stochastic Forcing, and its Applications

Ascione

Mishura

Pirozzi

2019

Methodol Comput Appl Probab

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We consider a fractional Ornstein-Uhlenbeck process involving a stochastic forcing term in the drift, as a solution of a linear stochastic differential equation driven by a fractional Brownian motion. For such process we specify mean and covariance functions, concentrating on their asymptotic behavior. This gives us a sort of short-or long-range dependence, under specified hypotheses on the covariance of the forcing process. Applications of this process in neuronal modeling are discussed, providing an example of a stochastic forcing term as a linear combination of Heaviside functions with random center. Simulation algorithms for the sample path of this process are finally given.

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Section: 42mentioning

confidence: 99%

Fractional Ornstein-Uhlenbeck Process with Stochastic Forcing, and its Applications

Ascione

Mishura

Pirozzi

2019

Methodol Comput Appl Probab

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“…Remark 4.6. For 1 > t > s from (13) and (14) it follows that E(Z t − Z s ) 2 exponentially tends to zero as s → 1. It can be shown that all derivatives with respect to s of this expectation also tend to zero as s → 1.…”

Section: Probability Of Hitting Zero By the Fractional Ornstein -Uhlementioning

confidence: 99%

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

Mishura

Piterbarg

Ralchenko

et al. 2019

Theor. Probability and Math. Statist.

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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.

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“…as T → ∞. Further, inserting (22) into (10), and applying the expansion (45) from Appendix A, we obtain…”

Section: Lemma 4 the Vector Of Main Components Of The Mle Has The Fomentioning

confidence: 99%

Maximum likelihood estimation in the non-ergodic fractional Vasicek model

Lohvinenko¹,

Ralchenko²

2019

Modern Stochastics: Theory and Applications

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We investigate the fractional Vasicek model described by the stochastic differential equation dXt = (α − βXt) dt + γ dB H t , X0 = x0, driven by the fractional Brownian motion B H with the known Hurst parameter H ∈ (1/2, 1). We study the maximum likelihood estimators for unknown parameters α and β in the non-ergodic case (when β < 0) for arbitrary x0 ∈ R, generalizing the result of Tanaka, Xiao and Yu (2019) for particular x0 = α/β, derive their asymptotic distributions and prove their asymptotic independence.

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Hypothesis testing of the drift parameter sign for fractional Ornstein–Uhlenbeck process

Cited by 16 publications

References 18 publications

Fractional Ornstein-Uhlenbeck Process with Stochastic Forcing, and its Applications

Fractional Ornstein-Uhlenbeck Process with Stochastic Forcing, and its Applications

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

Maximum likelihood estimation in the non-ergodic fractional Vasicek model

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