Intermittency of Superpositions of Ornstein–Uhlenbeck Type Processes

Grahovac, Danijel; Leonenko, Nikolai; Sikorskii, Alla; Tešnjak, Irena

doi:10.1007/s10955-016-1616-7

Cited by 16 publications

(30 citation statements)

References 42 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…[11]). When there are only finitely many OU type processes in the superposition, the mixing property remains valid and implies the convergence of the aggregated process to Brownian motion (see [22]). Problems arise when one considers an infinite superposition of OU type processes.…”

Section: Introductionmentioning

confidence: 99%

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

Grahovac¹,

Leonenko²,

Sikorskii³

et al. 2019

Bernoulli

Self Cite

View full text Add to dashboard Cite

show abstract

Section: Introductionmentioning

confidence: 99%

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

Grahovac¹,

Leonenko²,

Sikorskii³

et al. 2019

Bernoulli

Self Cite

View full text Add to dashboard Cite

show abstract

“…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.

View full text Add to dashboard Cite

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

show abstract

“…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: 82%

“…We will show that τ X * (1) = 1 − α 2 . Since τ X * (0) = 0, τ X * is convex function ((Grahovac et al 2016, Proposition 2.1)) passing through three collinear points: (0, 0), (1, 1 − α 2 ), 2, 2 1 − α 2 . By (Grahovac et al 2018, Lemma 3), τ X * is linear and τ X * (q) = 1 − α 2 q for q ≤ 2 which would complete the proof.…”

Section: Proofs Related To Intermittencymentioning

confidence: 99%

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

Grahovac

Leonenko

Taqqu

2019

Stochastic Processes and their Applications

Self Cite

View full text Add to dashboard Cite

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,

show abstract

Intermittency of Superpositions of Ornstein–Uhlenbeck Type Processes

Cited by 16 publications

References 42 publications

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

The unusual properties of aggregated superpositions of Ornstein–Uhlenbeck type 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

Contact Info

Product

Resources

About