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

Abstract

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

Joint temporal and contemporaneous aggregation of random-coefficient AR(1) processes with infinite variance

“…It is -stable, H -self-similar with , has stationary dependent increments, and is related to the integrated superposition of Ornstein–Uhlenbeck processes discussed by Barndorff-Nielsen [1]. See also [11]. The joint characteristic function of is given by for , , , .…”

Intermittency and infinite variance: the case of integrated supOU processes

“…• For the first term on the right hand side we use some parts of the proof of [20,Lemma 5.1]. From the integration formula for the stochastic integral, for any Λ-integrable function f on R + × R, one has (see [30])…”

“…The type of limit depends on whether the Gaussian component is present in (1.1) or not, on the behavior of π in (1.1) near the origin and on the growth of the Lévy measure µ in (1.1) near the origin (see [19] for details). In the infinite variance case, the limiting behavior is even more complex as the limit process may additionally depend on the regular variation index of the marginal distribution (see [20] for details). The limiting behavior of the integrated process has practical significance since supOU processes may be used as stochastic volatility models, see [1,10] and the references therein.…”

“…We focus here on the limiting behavior of moments and on the intermittency in the case where X(t) has infinite variance and show that we can have intermittency even in this case. To establish the rate of growth of moments we make use of the limit theorems established in [20]. The type of the limiting process depends heavily on the structure of the underlying supOU process.…”

“…Because of (4.16) we can writeκ L1 (ζ) = k(ζ)κ Sγ (γ 1/γ σ,ρ,0) (ζ),where k is slowly varying at zero such that k(ζ) ∼ k(1/ζ) as ζ → 0 and κ Sγ (γ 1/γ σ,ρ,0) is a cumulant function of a stable distribution as in(2.3). By[20, Eq. (34)] we have that…”