Superpositions and Products of Ornstein-Uhlenbeck Type Processes: Intermittency and Multifractality

$Superpositions and Products of Ornstein-Uhlenbeck Type Processes: Intermittency and Multifractality$ Superpositions and Products of Ornstein-Uhlenbeck Type Processes: Intermittency and Multifractality Nikolai N. Leonenko School of Mathematics Cardiff University The second conference on Ambit Fields and Related Topics Aarhus August 15 Abstract Superpositions of stationary processes of Ornstein-Uhlenbeck (supOU) type have been introduced by Barndorff-Nielsen. We consider the constructions producing processes with long-range dependence and infinitely divisible marginal distributions. We consider additive functionals of supOU processes that satisfy the property referred to as intermittency. We investigate the properties of multifractal products of supOU processes. We present the general conditions for the Lq convergence of cumulative processes and investigate their q-th order moments and Rényifunctions. These functions are nonlinear, hence displaying the multifractality of the processes. We also establish the corresponding scenarios for the limiting processes, such as log-normal, log-gamma, log-tempered stable or log-normal tempered stable scenarios. Acknowledgments Joint work with Denis Denisov, University of Manchester, UK Danijel Grahovac, University of Osijek, Croatia Alla Sikorskii, Michigan State University, USA Murad Taqqu, Boston University, USA OU type process I OU Type process is the unique strong solution of the SDE: dX (t) = −λX (t)dt + dZ(λt) where λ > 0, fZ(t)gt≥0 is a (non-decreasing, for this talk) Léviprocess, and an initial condition X0 is taken to be independent of Z(t). Note, in general Zt doesn't have to be a non-decreasing Lévyprocess. I For properties of OU type processes and their generalizations see Mandrekar & Rudiger (2007), Barndorff-Nielsen (2001), Barndorff-Nielsen & Stelzer (2011). I The a.s. unique solution is of the form: Z t −λt −λ(s−t) X (t) = e X0 + e dZ(λs); t ≥ 0: 0 Infinite superposition of OU type process (k) Let fX (t)gk≥1 be a sequence of independent OU type processes. Define an infinite superposition as: 1 X (k) X1(t) = X (t) k=1 Infinite superpositions are well defined under the following assumption: 1 1 X (k) X (k) (A): EX (t) < 1 and VarX (t) < 1: k=1 k=1 Infinite superposition of OU type process Assumption (B): (B1): The self decomposable distributions of X (k) have all moments and cumulants of all orders. (B2): The marginal distributions of X (k) are closed under convolution with respect to at least one parameter δk , and all cumulants are proportional to that parameter. Covariance structure of infinite supOU processes The covariance function is of the form 1 X (k) −λk t RX1 (t) = cov(X1(0); X1(t)) = Var(X (t))e ; k=1 (k) Assumption (B) implies that: Var(Xt ) = δk C; where C is a constant, reflects parameters of the marginals of X (k). For the −(1+2(1−H)) specific choice of parameters δk = k ; 1=2 < H < 1; and λk = λ/k; λ > 0; we get 1 X 1 R (t) = C e−λt=k : X1 2 k1+2(1−H) k=1 The covariance function RX1 (t) is not integrable for the chosen parameters δk and λk , i.e. the infinite supOU exhibits long-range dependence (LRD), as we show next. Covariance structure of infinite supOU processes For the infinite superposition of OU type processes that satisfy condition (B) and condition (A), the covariance function of X1(t), (k) −(1+2(1−H)) 1 with specific λ = λ/k and δk = k , 2 < H < 1, can be written as L(t) R (t) = ; t > 0 X1 t2(1−H) where function L is a slowly varying at infinity function. Observe that, for given H, the covariance function is not integrable at infinity, hence infinite supOU process is long-range dependent (LRD). Scaling function and intermittency For a process fY (t)gt≥0, denote q = supfq > 0 : EjY (t)jq < 1; 8tg and assume that for the function below limit exists and is finite for every q 2 [0; q). Then define a scaling function at point q 2 [0; q) as: log EjY (t)jq τ(q) = lim ; q 2 [0; q): t!1 log t We say a stochastic process fYt gt≥0 is intermittent if there exist two points p; r 2 (0; q) such that τ(p)=p < τ(r)=r. Note: Scaling function is convex and non-decreasing function. The function τ(q)=q is always non-decreasing, and what makes the process Yt intermittent is the existence of points of strict increase. Example of a non-intermittent sequence Slow moment growth: Denote the sum of positive, independent and identically distributed (iid) random variable with finite Pn q moments: Sn = i=1 ξi . Then its q-th moment grows as: E(Sn) q q ∼ n (Eξ1) , and the scaling function of Sn is of the form: q τ(q) = lim log jS(n)j = log n = lim (q log n+q log Eξ1)= log n = q: t!1 E t!1 Remark: Intuitively, the scaling function shows how q-th moment of the process fY (t)gt≥0 asymptotically behaves as a function of time: q τ(q) EjY (t)j ∼ t ; t ! 1: Intermittency as multifractality Theorem. Let fX1(t); t ≥ 0g be a non-Gaussian (discrete) supOU process such that the cumulant function κX (ζ) of the OU process fX (t); t ≥ 0g is analytic in the neighborhood of (1) (2) the origin and κX = 0 and κX 6= 0. If τY is the scaling function of [t] X Y = fY (t) = [X1(i) − EX1(i)]; t ≥ 0g; i=1 then for every q ≥ q∗ τY (q) = q − α; where q∗ is the smallest even integer greater than 2α, α = 2(1 − H): Thus Y is intermittent. Note that for self-similar processes Y τY (q) = qH; Moreover, the function τ (q) 2(1 − H) Y = 1 − = H(q) q q is strictly increasing in q : τ (2) τ (3) τ (q) τ (1) < Y < Y < ::: < Y < Y 2 3 q The term H(q) in the exponent of the asymptotic behavior of the q-th cumulant of Y (tN): (q) qH(q) κY (Nt) = CqL(N)[Nt] (1 + o(1)); N ! 1; that is supOU processes are intermittent. Thus, intremittency can be interpreted as special case of multifractality defined below. Multifractal products of stochastic processes Kahane (1985, 1987), Mennersalo, Norros and Reidi (2002): for an independent copies Λ0(s); :::; Λn(s) of the mother process Λ(s): t Z A(t) = lim [Λ0(s) ····Λn(s)]ds; t 2 T 2 + n!1 R 0 where fA(t); t ≥ 0g is maltifractal, that is for some non-linear function ζ(q); q 2 Q ⊆ R q ζ(q) EA (t) ∼ t ; for example (Kolmogorov's lognormal scenario): ζ(q) = −aq2 + (a + 1)q; a > 0: For a monofractal processes ζ(q) is a linear function, for example ζ(q) = qH; where H 2 (0; 1) is the Hurst exponent (FBM, α-stable motion, and α-stable subordinator and its inverse). Examples: 1) Binomial cascade: Λn(s) is constant on dyadic intervals; 2) Martingale de Mandelbrot: EΛn(s) = 1; 3) Stationary cascade: Λn(s) is stationary: i) conservation EΛn(s) = 1 ii) "self-similarity" d n Λn(s) = Λ1(sb ); b > 1: We consider the case Λ(t) = eX1(t)−cX ; where X1(t) is supOU stationary process with marginal distributions: Gaussian, Gamma; IG; NIG; TS; NTS; VG; ::: Product process The conditions A0-A000 yield (i) EΛb (t) = M(1) = 1; (i) 2 VarΛb (t) = M(2) − 1 = σΛ < 1; (i) (i) i Cov(Λb (t1); Λb (t2)) = M(1; 1; (t1 − t2)b ) − 1; b > 1: We define the finite product processes n ( n ) Y (i) X (i) i Λn(t) = Λb (t) = exp X (tb ) ; t 2 [0; 1]; i=0 i=0 and the cumulative processes Z t An(t) = Λn(s)ds; n = 0; 1; 2;:::; t 2 [0; 1]; 0 where X (i)(t); i = 0; :::; n; ::::; are independent copies of a stationary process X (t); t ≥ 0: Random measures We also consider the corresponding positive random measures defined on Borel sets B of R+ : Z µn(B) = Λn(s)ds; n = 0; 1; 2;::: B Kahane (1987) proved that the sequence of random measures µn converges weakly almost surely to a random measure µ. Moreover, given a finite or countable family of Borel sets Bj on R+, it holds that limn!1 µn(Bj ) = µ(Bj ) for all j with probability one: Random measures continued The almost sure convergence of An (t) in countably many points of R+ can be extended to all points in R+ if the limit process A (t) is almost surely continuous. In this case, limn!1 An(t) = A(t) with probability one for all t 2 R+: As noted in Kahane (1987), there are two extreme cases: (i) An(t) ! A(t) in L1 for each given t, in which case A(t) is not almost surely zero and and is said to be fully active (non-degenerate) on R+; (ii) An(1) converges to 0 almost surely, in which case A(t) is said to be degenerate on R+. Sufficient conditions for non-degeneracy and degeneracy in a general situation and relevant examples are provided in Kahane (1987). The condition for complete degeneracy is detailed in Theorem 3 of Kahane (1987). Conditions We introduce the following conditions: 0 A Let Λ(t); t 2 R+ = [0; 1); be a measurable, separable, strictly stationary, positive stochastic process with EΛ(t) = 1. We call this process the mother process and consider the following setting: A00 Let Λ(t) = Λ(i); i = 0; 1; ::: be independent copies of the (i) (i) mother process Λ; and Λb be the rescaled version of Λ : (i) d (i) i Λb (t) = Λ (tb ); t 2 R+; i = 0; 1; 2;:::; where the scaling parameter b > 1; and =d denotes equality in finite-dimensional distributions.

Load more