Multivariate generalized Ornstein–Uhlenbeck processes

De Haan and Karandikar (1989) [7] introduced generalized Ornstein–Uhlenbeck processes as one-dimensional processes _(Vt)t≥0${\left(V_t\right)}_{t\ge 0}$ which are basically characterized by the fact that for each h>0$h>0$ the equidistantly sampled process _(Vnh)n∈N0${\left(V_{nh}\right)}_{n\in {\mathbb{N}}_0}$ satisfies the random recurrence equation _Vnh=_{A(n−1)h,nhV(n−1)h}+_B(n−1)h,nh$V_{nh}=A_{(n-1)h,nh}{V}_{(n-1)h}+B_{(n-1)h,nh}$, n∈N$n\in \mathbb{N}$, where _{(A(n−1)h,nh,B(n−1)h,nh)n∈N}${(A_{(n-1)h,nh},B_{(n-1)h,nh})}_{n\in \mathbb{N}}$ is an i.i.d. sequence with positive _A0,h$A_{0,h}$ for each h>0$h>0$. We generalize this concept to a multivariate setting and use it to define multivariate generalized Ornstein–Uhlenbeck (MGOU) processes which occur to be characterized by a starting random variable and some Lévy process (X,Y)$(X,Y)$ in R^m×m×R^m${\mathbb{R}}^{m\times m}\times {\mathbb{R}}^m$. The stochastic differential equation an MGOU process satisfies is also derived. We further study invariant subspaces and irreducibility of the models generated by MGOU processes and use this to give necessary and sufficient conditions for the existence of strictly stationary MGOU processes under some extra conditions.     

Ergodic properties of generalized Ornstein–Uhlenbeck processes

We investigate ergodic properties of the solution of the SDE $d{V}_t=V_{t?}d{U}_t+d{L}_t$, where $(U,L)$ is a bivariate Lévy process. This class of processes includes the generalized Ornstein–Uhlenbeck processes. 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.     

On Ornstein–Uhlenbeck driven by Ornstein–Uhlenbeck processes

We investigate the asymptotic behavior of the maximum likelihood estimators of the unknown parameters of positive recurrent Ornstein–Uhlenbeck processes driven by Ornstein–Uhlenbeck processes.     

Regularity for semigroups of Ornstein–Uhlenbeck processes

Under mild conditions on the characteristic exponent or the symbol of Lévy process, we derive explicit estimates for L ^p (dx) → L ^q (dx) (1 ≤ p ≤ q ≤ ∞) norms of semigroups and their gradients of the associated Lévy driven Ornstein–Uhlenbeck process. Our result efficiently applies to the class of Lévy driven Ornstein–Uhlenbeck processes, where the asymptotic behaviour near infinity for the symbol of Lévy process is known.     

A Note on time regularity of generalized Ornstein–Uhlenbeck processes with cylindrical stable noise

A necessary and sufficient condition of càdlàg modification of Ornstein–Uhlenbeck process with cylindrical stable noise in a Hilbert space is given in this Note. Applying this result, some questions in Time irregularity of generalized Ornstein–Uhlenbeck processes [C. R. Acad. Sci. Paris, Ser. I 348 (2010) 273–276] and Structural properties of semilinear SPDEs driven by cylindrical stable process [Probab. Theory Related Fields 149 (2011) 97–137] are answered.     

Asymptotic results for finite superpositions of Ornstein–Uhlenbeck processes

A model of intermittency based on superposition of Lévy driven Ornstein–Uhlenbeck processes is studied in [6 Grahovac, D., Leonenko, N., Sikorskii, A., and Te?niak, I. 2016. Intermittency of superpositions of Ornstein–Uhlenbeck type processes. J. Stat. Phys. 165:390–408.[Crossref], [Web of Science ®] , [Google Scholar]]. In particular, as shown in Theorem 5.1 in that paper, finite superpositions obey a (sample path) central limit theorem under suitable hypotheses. In this paper we prove large (and moderate) deviation results associated with this central limit theorem.     

Large deviations for squared radial Ornstein–Uhlenbeck processes

In this paper, we state a large deviation principle (LDP) and sharp LDP for maximum likelihood estimators of drift coefficients of generalized squared radial Ornstein–Uhlenbeck processes. For that purpose, we present an LDP in a class of non-steep cases, where the Gärtner–Ellis theorem cannot be applied.     

Volterra-type Ornstein–Uhlenbeck processes in space and time

We propose a novel class of temporo-spatial Ornstein–Uhlenbeck processes as solutions to Lévy-driven Volterra equations with additive noise and multiplicative drift. After formulating conditions for the existence and uniqueness of solutions, we derive an explicit solution formula and discuss distributional properties such as stationarity, second-order structure and short versus long memory. Furthermore, we analyze in detail the path properties of the solution process. In particular, we introduce different notions of càdlàg paths in space and time and establish conditions for the existence of versions with these regularity properties. The theoretical results are accompanied by illustrative examples.     

Maximum likelihood estimation for multiscale Ornstein–Uhlenbeck processes

We study the problem of estimating the parameters of an Ornstein–Uhlenbeck (OU) process that is the coarse-grained limit of a multiscale system of OU processes, given data from the multiscale system. We consider both the averaging and homogenization cases and both drift and diffusion coefficients. By restricting ourselves to the OU system, we are able to substantially improve the results with strong modes of convergence, and provide some intuition of what to expect in the general case. In particular, in the homogenisation case we derive optimal rates of sub-sampling to minimize the estimation errors.     

Parameter estimation for fractional Ornstein–Uhlenbeck processes of general Hurst parameter

Statistical Inference for Stochastic Processes - This paper studies the least squares estimator (LSE) for the drift parameter of an Ornstein–Uhlenbeck process driven by fractional Brownian...     

First passage times of reflected Ornstein–Uhlenbeck processes with two-sided jumps

In this paper we consider the first passage problem for reflected jump-type Ornstein–Uhlenbeck processes with two-reflecting barriers. We calculate the explicit joint Laplace transform of the first passage time and the corresponding undershoot when the jumps follow a two-sided mixed exponential law. The method of contour integrals proposed by Jacobsen and Jensen (in Stoch. Process. Appl. 117: 1330–1356, 2007) is applied to obtain the explicit joint Laplace transform. Finally, a comparison concerning Laplace transforms between the reflected case and non-reflected case is presented by taking smooth-pasting conditions at reflecting barriers into account.     

Pseudo-Poissonian processes with stochastic intensity and a class of processes generalizing the Ornstein–Uhlenbeck process

The definition of pseudo-Poissonian processes is given in the famous monograph of William Feller (1971, Vol. II, Chapter X). The contemporary development of the theory of information flows generates new interest in the detailed analysis of behavior and characteristics of pseudo-Poissonian processes. Formally, a pseudo-Poissonian process is a Poissonian subordination of the mathematical time of an independent random sequence (the time randomization of a random sequence). We consider a sequence consisting of independent identically distributed random variables with second moments. In this case, pseudo-Poissonian processes do not have independent increments, but it is possible to calculate the autocovariance function, and it turns out that it exponentially decreases. Appropriately normed sums of independent copies of such pseudo-Poissonian processes tend to the Ornstein–Uhlenbeck process. A generalization of driving Poissonian processes to the case where the intensity is random is considered and it is shown that, under this generalization, the autocovariance function of the corresponding pseudo-Poissonian process is the Laplace transform of the distribution of that random intensity. Stochastic choice principles for the distribution of the random intensity are shortly discussed and they are illustrated by two detailed examples.     

Ornstein–Uhlenbeck processes in Hilbert space with non-Gaussian stochastic volatility

We propose a non-Gaussian operator-valued extension of the Barndorff-Nielsen and Shephard stochastic volatility dynamics, defined as the square-root of an operator-valued Ornstein–Uhlenbeck process with Lévy noise and bounded drift. We derive conditions for the positive definiteness of the Ornstein–Uhlenbeck process, where in particular we must restrict to operator-valued Lévy processes with “non-decreasing paths”. It turns out that the volatility model allows for an explicit calculation of its characteristic function, showing an affine structure. We introduce another Hilbert space-valued Ornstein–Uhlenbeck process with Wiener noise perturbed by this class of stochastic volatility dynamics. Under a strong commutativity condition between the covariance operator of the Wiener process and the stochastic volatility, we can derive an analytical expression for the characteristic functional of the Ornstein–Uhlenbeck process perturbed by stochastic volatility if the noises are independent. The case of operator-valued compound Poisson processes as driving noise in the volatility is discussed as a particular example of interest. We apply our results to futures prices in commodity markets, where we discuss our proposed stochastic volatility model in light of ambit fields.     

Greedy Approximation of High-Dimensional Ornstein–Uhlenbeck Operators

We investigate the convergence of a nonlinear approximation method introduced by Ammar et?al. (J. Non-Newtonian Fluid Mech. 139:153–176, 2006) for the numerical solution of high-dimensional Fokker–Planck equations featuring in Navier–Stokes–Fokker–Planck systems that arise in kinetic models of dilute polymers. In the case of Poisson’s equation on a rectangular domain in ?², subject to a homogeneous Dirichlet boundary condition, the mathematical analysis of the algorithm was carried out recently by Le Bris, Lelièvre and Maday (Const. Approx. 30:621–651, 2009), by exploiting its connection to greedy algorithms from nonlinear approximation theory, explored, for example, by DeVore and Temlyakov (Adv. Comput. Math. 5:173–187, 1996); hence, the variational version of the algorithm, based on the minimization of a sequence of Dirichlet energies, was shown to converge. Here, we extend the convergence analysis of the pure greedy and orthogonal greedy algorithms considered by Le Bris et al. to a technically more complicated situation, where the Laplace operator is replaced by an Ornstein–Uhlenbeck operator of the kind that appears in Fokker–Planck equations that arise in bead–spring chain type kinetic polymer models with finitely extensible nonlinear elastic potentials, posed on a high-dimensional Cartesian product configuration space D=D ₁×?×D _N contained in ? ^Nd , where each set D _i , i=1,…,N, is a bounded open ball in ? ^d , d=2,3.     

Parameter estimation for fractional Ornstein–Uhlenbeck processes at discrete observation

This paper deals with the problem of estimating the parameters for fractional Ornstein–Uhlenbeck processes from discrete observations when the Hurst parameter H is known. Both the drift and the diffusion coefficient estimators of discrete form are obtained based on approximating integrals via Riemann sums with Hurst parameter H ∈ (1/2, 3/4). By adapting the stochastic integral representation to the fractional Brownian motion, these two estimators can be efficiently computed by the use of computer software. Numerical examples are presented to examine the performance of our method. An application to real data is also presented to show how to apply this method in practice.     

Poissonian subordinators,the Wiener–Ornstein–Uhlenbeck field,and a relation between the Ornstein–Uhlenbeck processes and Brownian bridges

We apply to a sequence of i.i.d. random variables a time change operator via a Poisson process that is independent of this sequence. We consider sums of independent copies of processes constructed in this way and having continuous time. Finite limit distributions of these sums coincide with the finite limit distributions of the Wiener–Ornstein–Uhlenbeck field that is the tensor product of a Brownian motion and the Ornstein–Uhlenbeck process. The transition characteristics of the limit Ornstein–Uhlenbeck process are described by Brownian bridges that are builded into the Wiener–Ornstein–Uhlenbeck field. Bibliography: 4 titles.