We analyze {\em the Rosenblatt process} which is a selfsimilar process with stationary increments and which appears as limit in the so-called {\em Non Central Limit Theorem} (Dobrushin and Major (1979), Taqqu (1979)). This process is non-Gaussian and it lives in the second Wiener chaos. We give its representation as a Wiener-Itô multiple integral with respect to the Brownian motion on a finite interval and we develop a stochastic calculus with respect to it by using both pathwise type calculus and Malliavin calculus.
Let $Z^{α,β}$ be the Rosenblatt sheet with the representation $$ Z^{α,β}(t,s)=\int^t_0\int^s_0\int^t_0\int^s_0Q^α(t,y_1,y_2)Q^β(s,u_1,u_2)B(dy_1,du_1)B(dy_2,du_2) $$ where $B$ is a Brownian sheet, $\frac12<α,β<1$, $Q^α$ and $Q^β$ are the given kernel. In this paper, we contruct multiple Wiener integrals of the form \begin{align*} \int^t_0\int^s_0\int^t_0\int^s_0&[k_1(y_1,y_2)^{-\frac12α}(u_1,u_2)^{-\frac12β}+k_2(y_1\vee y_2)^{\frac12α}(y_1\wedge y_2)^{-\frac12α}|y_1-y_2|^{α-1}\\ &\cdot(u_1\vee u_2)^{\frac12β}(u_1\wedge u_2)^{-\frac12β}|u_1-u_2|^{β-1}]B(dy_1,du_1)B(dy_2,du_2),~~k_1,k_2\geq0, \e…
We consider multifractional process given by double Ito--Wiener integrals, which generalize the multifractional Rosenblatt process. We prove that this process is continuous and has a square integrable local time.
A stochastic calculus is given for processes described by stochastic integrals with respect to fractional Brownian motions and Rosenblatt processes somewhat analogous to the stochastic calculus for Itô processes. These processes for this stochastic calculus arise naturally from a stochastic chain rule for functionals of Rosenblatt processes; and some Itô-type expressions are given here. Furthermore, there is some analysis of these results for their applications to problems using Rosenblatt noise.
The Rosenblatt process is a self-similar non-Gaussian process which lives in second Wiener chaos, and occurs as the limit of correlated random sequences in so-called \textquotedblleft non-central limit theorems\textquotedblright. It shares the same covariance as fractional Brownian motion. We study the asymptotic distribution of the quadratic variations of the Rosenblatt process based on long filters, including filters based on high-order finite-difference and wavelet-based schemes. We find exact formulas for the limiting distributions, which we then use to devise strongly consistent estimator…
In this paper, we define a stochastic calculus with respect to the Rosenblatt process by means of white noise distribution theory. For this purpose, we compute the translated characteristic function of the Rosenblatt process at time $t>0$ in any direction $ξ\in S(\mathbb{R})$ and the derivative of the Rosenblatt process in the white noise sense. Using Wick multiplication by the former derivative and Pettis integration, we define our stochastic integral with respect to the Rosenblatt process for a wide class of distribution processes. We obtain an explicit formula for the variance of such a sto…
We consider the existence and Hölder continuity conditions for the self-intersection local time of Rosenblatt process. Moreover, we study the cases of intersection local time and collision local time, respectively.
The paper concerns the image, level and sojourn time sets associated with sample paths of the Rosenblatt process. We obtain results regarding the Hausdorff (both classical and macroscopic), packing and intermediate dimensions, and the logarithmic and pixel densities. As a preliminary step we also establish the time inversion property of the Rosenblatt process, as well as some technical points regarding the distribution of $Z$.
The Rosenblatt process was obtained by Taqqu (1975) from convergence in distribution of partial sums of strongly dependent random variables. In this paper we give a particle picture approach to the Rosenblatt process with the help of intersection local time and white noise analysis, and discuss measuring its long range dependence by means of a number called dependence exponent.
In this paper, we combine Hida distribution theory and Sobolev-Watanabe-Kree spaces in order to study finely the link between forward integrals obtained by regularization and Wick-Itô integrals with respect to fractional Brownian motion and the Rosenblatt process. The new methodology developed in this paper allows to retrieve results for fractional Brownian motion and to obtain new results regarding the Rosenblatt process. In particular, an Itô formula for functionals of the Rosenblatt process is obtained which holds in the space of square-integrable random variables.
We study the convergence in distribution, as $H\to \frac{1}{2}$ and as $H\to 1$, of the integral $\int_{\mathbb{R}} f(u) dZ^{H}(u) $, where $Z ^{H}$ is a Rosenblatt process with self-similarity index $H\in \left( \frac{1}{2}, 1\right) $ and $f$ is a suitable deterministic function. We focus our analysis on the case of the Rosenblatt Ornstein-Uhlenbeck process, which is the solution of the Langevin equation driven by the Rosenblatt process.
We provide a particle picture representation for the non-symmetric Rosenblatt process and for Hermite processes of any order, extending the result of Bojdecki, Gorostiza and Talarczyk in~\cite{FILT}. We show that these processes can be obtained as limits in the sense of finite-dimensional distributions of certain functionals of a system of particles evolving according to symmetric stable Lévy motions. In the case of $k$-Hermite processes the corresponding functional involves $k$-intersection local time of symmetric stable Lévy processes
We use techniques of Malliavin calculus to study the convergence in law of a family of generalized Rosenblatt processes $Z_γ$ with kernels defined by parameters $γ$ taking values in a tetrahedral region $Δ$ of $\RR^q$. We prove that, as $γ$ converges to a face of $Δ$, the process $Z_γ$ converges to a compound Gaussian distribution with random variance given by the square of a Rosenblatt process of one lower rank. The convergence in law is shown to be stable. This work generalizes a previous result of Bai and Taqqu, who proved the result in the case $q=2$ and without stability.
The generalized Rosenblatt process is obtained by replacing the single critical exponent characterizing the Rosenblatt process by two different exponents living in the interior of a triangular region. What happens to that generalized Rosenblatt process as these critical exponents approach the boundaries of the triangle? We show by two different methods that on each of the two symmetric boundaries, the limit is non-Gaussian. On the third boundary, the limit is Brownian motion. The rates of convergence to these boundaries are also given. The situation is particularly delicate as one approaches t…
We consider the class of all the Hermite processes $(Z_{t}^{(q,H)})_{t\in \lbrack 0,1]}$ of order $q\in \mathbf{N}^{\ast}$ and with Hurst parameter $% H\in (\frac{1}{2},1)$. The process $Z^{(q,H)}$ is $H$-selfsimilar, it has stationary increments and it exhibits long-range dependence identical to that of fractional Brownian motion (fBm). For $q=1$, $Z^{(1,H)}$ is fBm, which is Gaussian; for $q=2$, $Z^{(2,H)}$ is the Rosenblatt process, which lives in the second Wiener chaos; for any $q>2$, $Z^{(q,H)}$ is a process in the $q$th Wiener chaos. We study the variations of $Z^{(q,H)}$ for any $q$, b…
In this paper, we prove a Donsker type approximation theorem for the Rosenblatt process, which is a selfsimilar stochastic process exhibiting long range dependence. By using numerical results and simulated data, we show that this approximation performs very well. We use this result to construct a binary market model driven by this process and we show that the model admits arbitrage opportunities.
We prove that we can identify three types of pointwise behaviour in the regularity of the (generalized) Rosenblatt process. This extends to a non Gaussian setting previous results known for the (fractional) Brownian motion. On this purpose, fine bounds on the increments of the Rosenblatt process are needed. Our analysis is essentially based on various wavelet methods.
We study the least squares estimator for the drift parameter of the Langevin stochastic equation driven by the Rosenblatt process. Using the techniques of the Malliavin calculus and the stochastic integration with respect to the Rosenblatt process, we analyze the consistency and the asymptotic distribution of this estimator. We also introduce alternative estimators, which can be simulated, and we study their asymptotic properties.
We study persistence probabilities of Hermite processes. As a tool, we derive a general decorrelation inequality for the Rosenblatt process, which is reminiscent of Slepian's lemma for Gaussian processes or the FKG inequality and which may be of independent interest. This allows to compute the persistence exponent for the Rosenblatt process. For general Hermite processes, we derive upper and lower bounds for the persistence probabilites with the conjectured persistence exponent, but with non-matching boundaries.
By using chaos expansion into multiple stochastic integrals, we make a wavelet analysis of two self-similar stochastic processes: the fractional Brownian motion and the Rosenblatt process. We study the asymptotic behavior of the statistic based on the wavelet coefficients of these processes. Basically, when applied to a non-Gaussian process (such as the Rosenblatt process) this statistic satisfies a non-central limit theorem even when we increase the number of vanishing moments of the wavelet function. We apply our limit theorems to construct estimators for the self-similarity index and we ill…
Let $Z = (Z_t)_{t \geq 0}$ be the Rosenblatt process with Hurst index $H \in (1/2, 1)$. We prove joint continuity for the local time of $Z$, and establish Hölder conditions for the local time. These results are then used to study the irregularity of the sample paths of $Z$. Based on analogy with similar known results in the case of fractional Brownian motion, we believe our results are sharp. A main ingredient of our proof is a rather delicate spectral analysis of arbitrary linear combinations of integral operators, which arise from the representation of the Rosenblatt process as an element in…
The Rosenblatt distribution appears as limit in non-central limit theorems. The generalized Rosenblatt distribution is obtained by allowing different power exponents in the kernel that defines the usual Rosenblatt distribution. We derive an explicit formula for its third moment, correcting the one in \citet{maejima:tudor:2012:selfsimilar} and \citet{tudor:2013:analysis}. Evaluating this formula numerically, we are able to confirm that the class of generalized Hermite processes is strictly richer than the class of Hermite processes.
Let $(Z^{q, H}_t)_{t \in [0, 1]^d}$ denote a $d$-parameter Hermite random field of order $q \geq 1$ and self-similarity parameter $H = (H_1, \ldots, H_d) \in (\frac{1}{2}, 1)^d$. This process is $H$-self-similar, has stationary increments and exhibits long-range dependence. Particular examples include fractional Brownian motion ($q=1$, $d=1$), fractional Brownian sheet $(q=1, d \geq 2)$, Rosenblatt process ($q=2$, $d=1$) as well as Rosenblatt sheet $(q=2, d \geq 2)$. For any $q \geq 2, d\geq 1$ and $H \in (\frac{1}{2}, 1)^d$ we show in this paper that a proper normalization of the quadratic va…
Hermite processes are self--similar processes with stationary increments which appear as limits of normalized sums of random variables with long range dependence. The Hermite process of order $1$ is fractional Brownian motion and the Hermite process of order $2$ is the Rosenblatt process. We consider here the sum of two Hermite processes of order $q\geq 1$ and $q+1$ and of different Hurst parameters. We then study its quadratic variations at different scales. This is akin to a wavelet decomposition. We study both the cases where the Hermite processes are dependent and where they are independen…
We define Wiener integrals with respect to Yeh processes and study their properties. In particular, we obtain the martingale property of the associated stochastic processes and give a series expansion of Wiener integrals with respect to centered Yeh process. Moreover, we derive a representation of an Yeh process in terms of a random series.
A reduction theorem is proved for functionals of Gamma-correlated random fields with long-range dependence in d-dimensional space. In the particular case of a non-linear function of a chi-squared random field with Laguerre rank equal to one, we apply the Karhunen-Loéve expansion and the Fredholm determinant formula to obtain the characteristic function of its Rosenblatt-type limit distribution. When the Laguerre rank equals one and two, we obtain the multiple Wiener-Itô stochastic integral representation of the limit distribution. In both cases, an infinite series representation in terms of in…
We prove that, on the classical Wiener space, the random variable $\sup_{0\le t \le T} W_t$ admits a measure as second Malliavin derivative, whose total variation measure is finite and singular w.r.t.\ the Wiener measure.
We consider a class of measures absolutely continuous with respect to the distribution of the stopped Wiener process $w(\cdot\wedgeτ)$. Multiple stochastic integrals, that lead to the analogue of the Itô-Wiener expansions for such measures, are described. An analogue of the Krylov-Veretennikov formula for functionals $f=φ(w(τ))$ is obtained.
This paper deals with the study of the Malliavin calculus of Euclidean motions on Wiener space, (i.e. transformations induced by general measure preserving transformations, called `rotations', and H-valued shifts) and the associated flows on abstract Wiener spaces
We study sample path deviations of the Wiener process from three different representations of its bridge: anticipative version, integral representation and space-time transform. Although these representations of the Wiener bridge are equal in law, their sample path behavior is quite different. Our results nicely demonstrate this fact. We calculate and compare the expected absolute, quadratic and conditional quadratic path deviations of the different representations of the Wiener bridge from the original Wiener process. It is further shown that the presented qualitative behavior of sample path …
Given an R^d-valued supercritical branching Wiener process, let D(A,T) be the number of particles in a subset A of R^d at time T, (T=0,1,2,...). We provide a complete asymptotic expansion of D(A,T) as T goes to infinity, generalizing the work of X.Chen.
Wiener integral for the coordinate process is defined under the $ σ$-finite measure unifying Brownian penalisations, which has been introduced by Najnudel, Roynette and Yor. Its decomposition before and after last exit time from 0 is studied. This study prepares for the author's recent study of Cameron--Martin formula for the $ σ$-finite measure.
The problem of approximating/tracking the value of a Wiener process is considered. The discretization points are placed at times when the value of the process differs from the approximation by some amount, here denoted by eta. It is found that the limiting difference, as eta goes to 0, between the approximation and the value of the process normalized with eta converges in distribution to a triangularly distributed random variable.
We consider a massless tracer particle moving in a random, stationary, isotropic and divergence free velocity field. We identify a class of fields, for which the limit of the laws of appropriately scaled tracer trajectory processes is non-Gaussian but a Rosenblatt type of process.
In this work cylindrical Wiener processes on Banach spaces are defined by means of cylindrical stochastic processes, which are a well considered mathematical object. This approach allows a definition which is a simple straightforward extension of the real-valued situation. We apply this definition to introduce a stochastic integral with respect to cylindrical Wiener processes. Again, this definition is a straightforward extension of the real-valued situation which results now in simple conditions on the integrand. In particular, we do not have to put any geometric constraints on the Banach spa…
Let $G$ be a non--linear function of a Gaussian process $\{X_t\}_{t\in\mathbb{Z}}$ with long--range dependence. The resulting process $\{G(X_t)\}_{t\in\mathbb{Z}}$ is not Gaussian when $G$ is not linear. We consider random wavelet coefficients associated with $\{G(X_t)\}_{t\in\mathbb{Z}}$ and the corresponding wavelet scalogram which is the average of squares of wavelet coefficients over locations. We obtain the asymptotic behavior of the scalogram as the number of observations and scales tend to infinity. It is known that when $G$ is a Hermite polynomial of any order, then the limit is either…
We consider a class of Markov processes with resettings, where at random times, the Markov processes are restarted from a predetermined point or a region. These processes are frequently applied in physics, chemistry, biology, economics, and in population dynamics. In this paper we establish the local large deviation principle (LLDP) for the Wiener processes with random resettings, where the resettings occur at the arrival time of a Poisson process. Here, at each resetting time, a new resetting point is selected at random, according to a conditional distribution.
We examine the asymptotic behaviour of the sample autocovariance in a continuous-time moving average model with long-range dependence. We show that it is either asymptotically Rosenblatt distributed or stable distributed. This shows that results by Horváth and Kokoszka for discrete-time moving average processes with long memory also hold for continuous-time moving average processes.
We consider a mixed stochastic differential equation driven by possibly dependent fractional Brownian motion and Brownian motion. Under mild regularity assumptions on the coefficients, it is proved that the equation has a unique solution.
Fourier-Wiener transform of the formal expression for multiple self-intersection local time is described in terms of the integral, which is divergent on the diagonals. The method of regularization we use in this work related to regularization of functions with non-integrable singularities. The strong local nondeterminism property, which is more restrictive than the property of local nondeterminism introduced by S.Berman is considered. Its geometrical meaning in the construction of the regularization is investigated. As the example the problem of regularization is solved for the compact perturb…
We derive the Lévy-Khintchine representation of the Wiener-Hopf factors for the Normal Inverse Gaussian (NIG) process as well as a representation which is similar to the moment generating function (MGF) of a generalized gamma convolution (GGC). We show, via this representation, that for some parameters the Wiener-Hopf factors are, in fact, the MGFs of GGCs. Further, we develop two seperate methods of approximating the Wiener-Hopf factors, both based on Padé approximations of their Taylor series expansions; we show how the latter may be calculated exactly to any order. The first approximation y…
Let $ξ(k,n)$ be the local time of a simple symmetric random walk on the line. We give a strong approximation of the centered local time process $ξ(k,n)-ξ(0,n)$ in terms of a Wiener sheet and an independent Wiener process, time changed by an independent Brownian local time. Some related results and consequences are also established.
Let us consider the process $(X_t^{(α)})_{t\in[0,T)}$ given by the SDE $dX_t^{(α)} = -\fracα{T-t}X_t^{(α)} dt+ dB_t$, $t\in[0,T)$, where $α\in R$, $T\in(0,\infty)$, and $(B_t)_{t\geq 0}$ is a standard Wiener process. In case of $α>0$ the process $X^{(α)}$ is known as an $α$-Wiener bridge, in case of $α=1$ as the usual Wiener bridge. We prove that for all $α,β\in R$, $α\neβ$, the probability measures induced by the processes $X^{(α)}$ and $X^{(β)}$ are singular on C[0,T). Further, we investigate regularity properties of $X_t^{(α)}$ as $t\uparrow T$.
We introduce the notion of {\em covariance measure structure} for square integrable stochastic processes. We define Wiener integral, we develop a suitable formalism for stochastic calculus of variations and we make Gaussian assumptions only when necessary. Our main examples are finite quadratric variation processes with stationary increments and the bifractional Brownian motion.
We point out an easy link between two striking identities on exponential functionals of the Wiener process and the Wiener bridge originated by Bougerol, and Donati-Martin, Matsumoto and Yor, respectively. The link is established using a continuous one-parameter family of Gaussian processes known as $α$-Wiener bridges or scaled Wiener bridges, which in case $α=0$ coincides with a Wiener process and for $α=1$ is a version of the Wiener bridge.
We derive formulae for some ratios of the Macdonald functions, which are simpler and easier to treat than known formulae. The result gives two applications in probability theory. One is the formula for the L{é}vy measure of the distribution of the first hitting time of a Bessel process and the other is an explicit form for the expected volume of the Wiener sausage for an even dimensional Brownian motion. Moreover, the result enables us to write down the algebraic equations whose roots are the zeros of Macdonald functions.
In this paper, we define, via Fourier transform, an ergodic flow of transformations of a Wiener space which preserves the law of the Ornstein-Uhlenbeck process and which interpolates the iterations of a transformation previously defined by Jeulin and Yor. Then, we give a more explicit expression for this flow, and we construct from it a continuous gaussian process indexed by R^2, such that all its restriction obtained by fixing the first coordinate are Ornstein-Uhlenbeck processes.
We collect, scattered through literature, as well as we prove some new properties of two Markov processes that in many ways resemble Wiener and Ornstein--Uhlenbeck processes. Although processes considered in this paper were defined either in non-commutative probability context or through quadratic harnesses we define them once more as so to say 'continuous time ' generalization of a simple, symmetric, discrete time process satisfying simple conditions imposed on the form of its first two conditional moments. The finite dimensional distributions of the first one (say X=(X_{t})_{t\geq0} called q…
An elementary construction of the Wiener process is discussed, based on a proper sequence of simple symmetric random walks that uniformly converge on bounded intervals, with probability 1. This method is a simplification of F.B. Knight's and P. Révész's. The same sequence is applied to give elementary (Lebesgue-type) definitions of Itô and Stratonovich sense stochastic integrals and to prove the basic Itô formula. The resulting approximating sums converge with probability 1. As a by-product, new elementary proofs are given for some properties of the Wiener process, like the almost sure non-dif…
The inverse first-passage problem for a Wiener process $(W_t)_{t\ge0}$ seeks to determine a function $b{}:{}\mathbb{R}_+\to\mathbb{R}$ such that \[τ=\inf\{t>0| W_t\ge b(t)\}\] has a given law. In this paper two methods for approximating the unknown function $b$ are presented. The errors of the two methods are studied. A set of examples illustrates the methods. Possible applications are enlighted.
In this paper we introduce a ten-parameter family of Lévy processes for which we obtain Wiener-Hopf factors and distribution of the supremum process in semi-explicit form. This family allows an arbitrary behavior of small jumps and includes processes similar to the generalized tempered stable, KoBoL and CGMY processes. Analytically it is characterized by the property that the characteristic exponent is a meromorphic function, expressed in terms of beta and digamma functions. We prove that the Wiener-Hopf factors can be expressed as infinite products over roots of a certain transcendental equat…
