Isotropic Brownian flows (IBFs) are a fairly natural class of stochastic flows which has been studied extensively by various authors. Their rich structure allows for explicit calculations in several situations and makes them a natural object to start with if one wants to study more general stochastic flows. Often the intuition gained by understanding the problem in the context of IBFs transfers to more general situations. However, the obvious link between stochastic flows, random dynamical systems and ergodic theory cannot be exploited in its full strength as the IBF does not have an invariant probability measure but rather an infinite one. Isotropic Ornstein-Uhlenbeck flows are in a sense localized IBFs and do have an invariant probability measure. The imposed linear drift destroys the translation invariance of the IBF, but many other important structure properties like the Markov property of the distance process remain valid and allow for explicit calculations in certain situations. The fact that isotropic Ornstein-Uhlenbeck flows have invariant probability measures allows one to apply techniques from random dynamical systems theory. We demonstrate this by applying the results of Ledrappier and Young to calculate the Hausdorff dimension of the statistical equilibrium of an isotropic Ornstein-Uhlenbeck flow.
For an Ornstein-Uhlenbeck process driven by a double exponential jump diffusion process, we obtain formulas for the joint Laplace transform of it and its occupation times. The approach used is remarkable and can be extended to investigate the occupation times of an Ornstein-Uhlenbeck process driven by a more general Levy process.
We investigate the asymptotic properties of the minimum $L_1$-norm estimator of the drift parameter for fractional Ornstein-Uhlenbeck type process driven by a general Gaussian process.
We obtain a Berry-Esseen type bound for the distribution of the maximum likelihood estimator of the drift parameter for fractional Ornstein-uhlenbeck type process driven by sub-fractional Brownian motion.
We define a time-changed fractional Ornstein-Uhlenbeck process by composing a fractional Ornstein-Uhlenbeck process with the inverse of a subordinator. Properties of the moments of such process are investigated and the existence of the density is shown. We also provide a generalized Fokker-Planck equation for the density of the process.
In this article, we introduce a non Gaussian long memory process constructed by the aggregation of independent copies of a fractional Lévy Ornstein-Uhlenbeck process with random coefficients. Several properties and a limit theorem are studied for this new process. Finally, some simulations of the limit process are shown.
The paper studies a class of Ornstein-Uhlenbeck processes on the classical Wiener space. These processes are associated with a diffusion type Dirichlet form whose corresponding diffusion operator is unbounded in the Cameron-Martin space. It is shown that the distributions of certain finite dimensional Ornstein-Uhlenbeck processes converge weakly to the distribution of such an infinite dimensional Ornstein-Uhlenbeck process. For the infinite dimensional processes, the ordinary scalar quadratic variation is calculated. Moreover, relative to the stochastic calculus via regularization, the scalar …
Markovian intertwining relations between two Markov semigroups are related to the partial inclusion of the spectra of their generators, at least for finite ergodic processes. We check the limitations of this observation by investigating the Markov intertwining relations between the Ehrenfest, Yule and Ornstein-Uhlenbeck processes, whose spectra are all included into -$\mathbb{Z}_+$. As a by-product, we offer a clarification of an intertwining relation found in Biane [2] between the Yule and the Ornstein-Uhlenbeck processes.
For an Ornstein-Uhlenbeck process driven by fractional Brownian motion with Hurst index $H\in [\frac12,\frac34]$, we show the Berry-Esséen bound of the least squares estimator of the drift parameter. We use an approach based on Malliavin calculus given by Kim and Park \cite{kim 3}.
We investigate ergodic properties of generalized Ornstein--Uhlenbeck processes. In particular, we provide sufficient conditions for ergodicity, and for subexponential and exponential convergence to the invariant probability measure. We use the Foster--Lyapunov method. The drift conditions are obtained using the explicit form of the generator of the continuous process. In some special cases the optimality of our results can be shown.
We continue the investigation of sample paths of $q$-Ornstein-Uhlenbeck process. We show that for all $q\in(-1,1)$, the process has big jumps crossing from near one end point of the domain to the other with positive probability. Moreover, the number of such jumps in an appropriately enlarged window converges weakly to a Poisson random variable.
In this article, existence of the $k$-th order derivatives of local time $ \widehatα^{(k)}(x,t)$ is considered for two d-dimensional fractional Ornstein-Uhlenbeck processes $X^{H_1}_t$ and $\widetilde{X}^{H_2}_s$ with Hurst parameters $H_1$ and $H_2$, respectively. Moreover, H$\hat{o}$lder regularity condition of fractional Ornstein-Uhlenbeck process $X^{H}_t$ of local time $\tildeα^{(k)}(x,t)$ is obtained by some techniques using in Guo et al. (2017) and in Lou et al. (2017).
We give some examples of random fields that can be represented as space-domain scaled stationary Ornstein-Uhlenbeck fields defined on the plane. Namely, we study a tied-down Wiener bridge, tied-down scaled Wiener bridges, a Kiefer process and so called (F,G)-Wiener bridges.
In this paper, we consider the measure determined by a fractional Ornstein-Uhlenbeck process. For such measure, we establish a martingale representation theorem and consequently obtain the Logarithmic-Sobolev inequality. To this end, we also present the integration by parts formula for such measure, which is obtained via its pull back formula and the Bismut method.
We show that for any fixed $θ\in(-\fracπ{2},\,0)\cup (0,\,\fracπ{2})$, the 1-dimensional complex Ornstein-Uhlenbeck operator \begin{equation*} \tilde{\mathcal{L}}_θ= 4\cosθ\frac{\partial^2}{\partial z\partial \bar{z}}-e^{\miθ} z \frac{\partial}{\partial z}-e^{-\miθ}\bar{z} \frac{\partial}{\partial \bar{z}}, \end{equation*} is a normal (but nonsymmetric) diffusion operator.
In this paper, we study Ornstein-Uhlenbeck processes with Markov modulation, whose parameters depend on an external underlying two-state Markov process. Conditional mean and variance of such processes under given modulation are investigated from the point of view of the first passage probabilities and invariant measures. It is also studied the limiting behaviour under scaling conditions similar to Kac's scaling.
We investigate large deviation properties of the maximum likelihood drift parameter estimator for Ornstein--Uhlenbeck process driven by mixed fractional Brownian motion.
In this work we investigate the long time behavior of the Ornstein-Uhlenbeck process driven by Levy noise with regime-switching. We provide explicit criteria on the transience and recurrence of this process. Contrasted with the Ornstein-Uhlenbeck process driven simply by Brownian motion, whose stationary distribution must be light-tailed, both the jumps caused by the Levy noise and regime-switching described by Markov chain can derive the heavy-tailed property of the stationary distribution. In this work, the different role played by Levy measure and regime-switching process is clearly charact…
We give an explicit representation for the transition law of a tempered stable Ornstein-Uhlenbeck process and use it to develop a rejection sampling algorithm for exact simulation of increments from this process. Our results apply to general classes of both univariate and multivariate tempered stable distributions and contain a number of previously studied results as special cases.
Superpositions of Ornstein-Uhlenbeck type (supOU) processes form a rich class of stationary processes with a flexible dependence structure. The asymptotic behavior of the integrated and partial sum supOU processes can be, however, unusual. Their cumulants and moments turn out to have an unexpected rate of growth. We identify the property of fast growth of moments or cumulants as intermittency.
In this paper we study some convergence results concerning the one-dimensional distribution of a time-changed fractional Ornstein-Uhlenbeck process. In particular, we establish that, despite the time change, the process admits a Gaussian limit random variable. On the other hand, we prove that the process converges towards the time-changed Ornstein-Uhlenbeck as the Hurst index $H \to 1/2^+$, with locally uniform convergence of one-dimensional distributions. Moreover, we also achieve convergence in the Skorohod $J_1$-topology of the time-changed fractional Ornstein-Uhlenbeck process as $H \to 1/…
In order to approximate the exit time of a one-dimensional diffusion process, we propose an algorithm based on a random walk. Such an algorithm so-called Walk on Moving Spheres was already introduced in the Brownian context. The aim is therefore to generalize this numerical approach to the Ornstein-Uhlenbeck process and to describe the efficiency of the method.
The $q$-Ornstein-Uhlenbeck processes, $q\in(-1,1)$, are a family of stationary Markov processes that converge weakly to the standard Ornstein-Uhlenbeck process as $q$ tends to 1. It has been noticed recently that in terms of path properties, however, for each $q$ fixed the $q$-Ornstein-Uhlenbeck process behaves qualitatively different from their Gaussian counterpart in several aspects. Here, two limit theorems on the extremes of $q$-Ornstein-Uhlenbeck processes are established. Both results are based on the weak convergence of the tangent process at the lower boundary, a positive self-similar …
This paper addresses the problem of estimating drift parameter of the Ornstein - Uhlenbeck type process, driven by the sum of independent standard and fractional Brownian motions. The maximum likelihood estimator is shown to be consistent and asymptotically normal in the large-sample limit, using some recent results on the canonical representation and spectral structure of mixed processes.
In this paper we study some properties of the generalized Fokker-Planck equation induced by the time-changed fractional Ornstein-Uhlenbeck process. First of all, we exploit some sufficient conditions to show that a mild solution of such equation is actually a classical solution. Then we discuss an isolation result for mild solutions. Finally, we prove the weak maximum principle for strong solutions of the aforementioned equation and then a uniqueness result.
The maximal inequalities for diffusion processes have drawn increasing attention in recent years. However, the existing proof of the $L^p$ maximum inequalities for the Ornstein-Uhlenbeck process was dubious. Here we give a rigorous proof of the moderate maximum inequalities for the Ornstein-Uhlenbeck process, which include the $L^p$ maximum inequalities as special cases and generalize the remarkable $L^1$ maximum inequalities obtained by Graversen and Peskir [P. Am. Math. Soc., 128(10):3035-3041, 2000]. As a corollary, we also obtain a new moderate maximal inequality for continuous local marti…
In this paper, we will first give the numerical simulation of the sub-fractional Brownian motion through the relation of fractional Brownian motion instead of its representation of random walk. In order to verify the rationality of this simulation, we propose a practical estimator associated with the LSE of the drift parameter of mixed sub-fractional Ornstein-Uhlenbeck process, and illustrate the asymptotical properties according to our method of simulation when the Hurst parameter $H>1/2$.
We consider the fractional Ornstein-Uhlenbeck process with an unknown drift parameter and known Hurst parameter $H$. We propose a new method to test the hypothesis of the sign of the parameter and prove the consistency of the test. Contrary to the previous works, our approach is applicable for all $H\in(0,1)$. We also study the estimators for drift parameter for continuous and discrete observations and prove their strong consistency for all $H\in(0,1)$.
We present a class of Gauss-Markov processes which can be represented as space-time scaled stationary Ornstein-Uhlenbeck processes defined on the real line. We give several explicit examples of the representation for certain Gauss bridge processes. As an application, we derive a formula for the density function of the supremum location of certain standardized Gauss-Markov processes on compact time intervals. We also present some sufficient conditions under which mean centered Gauss-Markov processes take zero at a fixed time with probability one.
Ornstein-Uhlenbeck process of bounded variation is introduced as a solution of an analogue of the Langevin equation with an integrated telegraph process replacing a Brownian motion. There is an interval $I$ such that the process starting from the internal point of $I$ always remains within $I$. Starting outside, this process a. s. reaches this interval in a finite time. The distribution of the time for which the process falls into this interval is obtained explicitly. The certain formulae for the mean and the variance of this process are obtained on the basis of the joint distribution of the t…
