The unusual properties of aggregated superpositions of Ornstein–Uhlenbeck type processes

Grahovac, Danijel; Leonenko, Nikolai; Sikorskii, Alla; Taqqu, Murad S.

doi:10.3150/18-bej1044

Cited by 13 publications

(30 citation statements)

References 44 publications

(101 reference statements)

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“…Moreover for the MMA stochastic volatility models, we show the θ-weak dependence of the return process and the distributional limit of its sample moments. In [29,30,31], the limiting behavior of integrated and partial sums of supOU processes is analyzed in relation to the growth rate of their moments, called intermittency when the grow rate is fast. This leads to some conclusions regarding their asymptotic finite dimensional distributions and to identify different limiting theorems depending on the short or long memory shown by the supOU process.…”

Section: Introductionmentioning

confidence: 99%

Weak dependence and GMM estimation of supOU and mixed moving average processes

Curato¹,

Stelzer²

2019

Electron. J. Statist.

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We consider a mixed moving average (MMA) process X driven by a Lévy basis and prove that it is weakly dependent with rates computable in terms of the moving average kernel and the characteristic quadruple of the Lévy basis. Using this property, we show conditions ensuring that sample mean and autocovariances of X have a limiting normal distribution. We extend these results to stochastic volatility models and then investigate a Generalized Method of Moments estimator for the supOU process and the supOU stochastic volatility model after choosing a suitable distribution for the mean reversion parameter. For these estimators, we analyze the asymptotic behavior in detail.An important example of MMA are the supOU processes studied in [3,10,25,27]. In the univariate case, assume |x|>1 log(|x|) ν(dx) < ∞ and R − − 1 A π(dA) < ∞, where ν is a Lévy measure and π is the probability distribution on R − of the random parameter A, see Definition 2.1 for details. If Λ is a Lévy basis on R with those characteristics, then the processis called a supOU process. Whereas a non-Gaussian Ornstein-Uhlenbeck process necessarily exhibits autocorrelation e ah for h ∈ N, the supOU process has a flexible dependence structure. For example, its autocorrelations can show a polynomial decay depending on the probability distribution π. Moreover, when a discrete probability distribution π for the random parameter A is considered, we obtain a popular model used, for example, in stochastic volatility models [3], in modeling fractal activity times [36,37] and in astrophysics [35].MMA processes can also be used, under suitable conditions, as building blocks for more complex models. We study in this paper the class of MMA stochastic volatility models. An example of the class is the supOU SV model, defined in [10,11], where the log-price process (of some financial asset) is defined for t ∈ R + as

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Section: Introductionmentioning

confidence: 99%

Weak dependence and GMM estimation of supOU and mixed moving average processes

Curato¹,

Stelzer²

2019

Electron. J. Statist.

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“…Non-central limit theorems with convergence to fractional Brownian motion appeared in Barndorff-Nielsen & Leonenko (2005a), Leonenko & Taufer (2005). From the results presented here, it is now clear that these do not hold in general and that they depend on the rate of growth of the moments of the integrated process X * , see also Grahovac et al (2018), Grahovac et al (2016). We focus here on how an unusual rate of growth of the integrated process X * (t) can affect limit theorems.…”

Section: Introductionmentioning

confidence: 83%

Limit theorems, scaling of moments and intermittency for integrated finite variance supOU processes

Grahovac

Leonenko

Taqqu

2019

Stochastic Processes and their Applications

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Superpositions of Ornstein-Uhlenbeck type (supOU) processes provide a rich class of stationary stochastic processes for which the marginal distribution and the dependence structure may be modeled independently. We show that they can also display intermittency, a phenomenon affecting the rate of growth of moments. To do so, we investigate the limiting behavior of integrated supOU processes with finite variance. After suitable normalization four different limiting processes may arise depending on the decay of the correlation function and on the characteristic triplet of the marginal distribution.To show that supOU processes may exhibit intermittency, we establish the rate of growth of moments for each of the four limiting scenarios. The rate change indicates that there is intermittency, which is expressed here as a change-point in the asymptotic behavior of the absolute moments.holds with convergence in the sense of convergence of all finite dimensional distributions as T → ∞, then Z is H-self-similar for some H > 0, that is, for any constant c > 0, the finite dimensional distributions of Z(ct) are the same as those of c H Z(t). Brownian motion for example is self-similar with H = 1/2. Moreover, the normalizing sequence is of the form A T = ℓ(T )T H for some ℓ slowly varying at infinity. For self-similar process, the moments evolve as a power function of time since E|Z(t)| q = E|Z(1)| q t Hq and therefore the scaling function of Z is τ Z (q) = Hq. Hence (4) does not hold for self-similar processes. But it may not hold either for the process X * in (5) because one would expect that E|X * (T t)| q A q T → E|Z(t)| q , ∀t ≥ 0,

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“…On the other hand, the probability distribution π affects the dependence structure. See Barndorff-Nielsen (2001), Barndorff-Nielsen & Stelzer (2011), Barndorff-Nielsen & Stelzer (2013), Barndorff-Nielsen & Veraart (2013), Barndorff-Nielsen et al (2018), Grahovac, Leonenko, Sikorskii & Taqqu (2019) for details. SupOU processes provide models with analytically and stochastically tractable dependence structure displaying either weak or strong dependence and also having marginal distributions that are infinitely divisible.…”

Section: Introductionmentioning

confidence: 99%

“…In particular, q → τ Y (q)/q would be constant over q values for which (4) holds. It has been showed in Grahovac, Leonenko, Sikorskii & Taqqu (2019) that the integrated supOU process X * may have a scaling function which does not correspond to some self-similar process, namely τ X * (q) = q − α for a certain range of q. This happens, in particular, for a non-Gaussian integrated supOU process with marginal distribution having exponentially decaying tails and probability measure π in (1) regularly varying at zero.…”

Section: Introductionmentioning

confidence: 99%

“…Hence, intermittency implies the convergence of moments (4) must fail to hold beyond some critical value of q. See Grahovac et al (2016), Grahovac, Leonenko, Sikorskii & Taqqu (2019), which provide a complete picture on the behavior of moments in the case where X * (t) has finite variance. Intermittency refers in general to an unusual moment behavior.…”

Section: Introductionmentioning

confidence: 99%

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The Multifaceted Behavior of Integrated supOU Processes: The Infinite Variance Case

2019

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SupOU processes are superpositions of Ornstein-Uhlenbeck type processes with a random intensity parameter. They are stationary processes whose marginal distribution and dependence structure can be specified independently. Integrated supOU processes have then stationary increments and satisfy central and non-central limit theorems. Their moments, however, can display an unusual behavior known as "intermittency". We show here that intermittency can also appear when the processes have a heavy tailed marginal distribution and, in particular, an infinite variance.1 By aggregating the supOU process {X(t), t ∈ R} one obtains the integrated supOU process X * (t) = t 0

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The unusual properties of aggregated superpositions of Ornstein–Uhlenbeck type processes

Cited by 13 publications

References 44 publications

Weak dependence and GMM estimation of supOU and mixed moving average processes

Weak dependence and GMM estimation of supOU and mixed moving average processes

Limit theorems, scaling of moments and intermittency for integrated finite variance supOU processes

The Multifaceted Behavior of Integrated supOU Processes: The Infinite Variance Case

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