Generalized Ornstein-Uhlenbeck Process Research Articles

Continuity properties and infinite divisibility of stationary distributions of some generalized Ornstein–Uhlenbeck processes

Properties of the law μ of the integral ∫0∞c−Nt− dYt are studied, where c>1 and {(Nt, Yt), t≥0} is a bivariate Lévy process such that {Nt} and {Yt} are Poisson processes with parameters a and b, respectively. This is the stationary distribution of some generalized Ornstein–Uhlenbeck process. The law μ is parametrized by c, q and r, where p=1−q−r, q, and r are the normalized Lévy measure of {(Nt, Yt)} at the points (1, 0), (0, 1) and (1, 1), respectively. It is shown that, under the condition that p>0 and q>0, μc, q, r is infinitely divisible if and only if r≤pq. The infinite divisibility of the symmetrization of μ is also characterized. The law μ is either continuous-singular or absolutely continuous, unless r=1. It is shown that if c is in the set of Pisot–Vijayaraghavan numbers, which includes all integers bigger than 1, then μ is continuous-singular under the condition q>0. On the other hand, for Lebesgue almost every c>1, there are positive constants C1 and C2 such that μ is absolutely continuous whenever q≥C1p≥C2r. For any c>1 there is a positive constant C3 such that μ is continuous-singular whenever q>0 and max {q, r}≤C3p. Here, if {Nt} and {Yt} are independent, then r=0 and q=b/(a+b).

Reliability of linear systems under Poisson white noise

A method is developed for calculating the failure probability of a linear dynamic system subjected to Poisson white noise assumed to function satisfactorily if its state X ( t ) does not leave a safe set D during a specified time interval [ 0 , τ ] . The method (1) is based on properties of Poisson white noise, features of system response to Poisson white noise, and conditional Monte Carlo simulation, and (2) delivers upper bounds on system failure probability, up to estimation errors. Numerical results illustrating the application and the accuracy of the proposed method include the largest value distribution of a generalized Ornstein–Uhlenbeck process and the reliability of a simple linear oscillator subjected to Poisson white noise. It is also shown that the proposed method can be applied to assess the performance of multidegree of freedom linear systems and nonlinear systems subjected to Poisson white noise.

Scaling Limits for Gradient Systems in Random Environment

For interacting particle systems that satisfies the gradient condition, the hydrodynamic limit and the equilibrium fluctuations are well known. We prove that under the presence of a symmetric random environment, these scaling limits also hold for almost every choice of the environment, with homogenized coefficients that does not depend on the particular realization of the random environment.

The stationarity of multidimensional generalized Ornstein–Uhlenbeck processes

The multidimensional generalized Ornstein–Uhlenbeck process is defined as an extended version of the generalized Ornstein–Uhlenbeck process of which integral is with respect to a multidimensional Lévy process. The condition for the stationarity of the process and that for the convergence are respectively studied. The relationship between these two conditions is also clarified.

Eigenfunctions and matrix elements for a class of eigenvalue problems with staggered ladder spectra

We present an alternate solution to an eigenvalue problem which arises in the study of the Fokker-Planck equation for generalized Ornstein-Uhlenbeck processes. We obtain the staggered ladder spectra found previously but in addition we obtain the normalized eigenfunctions in terms of associated Laguerre polynomials. The representation of the eigenfunctions in this form greatly simplifies the evaluation of matrix elements required in calculating ensemble averages and correlation coefficients for various observables. The solution to the eigenvalue problem is given in the generic and general cases.

Scaling limit of fluctuations for the equilibrium Glauber dynamics in continuum

The Glauber dynamics investigated in this paper are spatial birth and death processes in a continuous system having a grand canonical Gibbs measure of Ruelle type as an invariant measure. We prove that such processes, when appropriately scaled, have as scaling limit a generalized Ornstein–Uhlenbeck process. First we prove convergence of the corresponding Dirichlet forms. This convergence requires only very weak assumptions. The interaction potential ϕ only has to be stable (S), integrable (I), and we have to assume the low activity high temperature regime. Under a slightly stronger integrability condition ( I ∞ ) and a conjecture on the Percus–Yevick equation we even can prove strong convergence of the corresponding generators. Finally, we prove that the scaled processes converge in law. Here the hardest part is to show tightness of the scaled processes (note that the processes only have càdlàg sample path). For the proof we have to assume that the interaction potential is positive (P). The limiting process then is identified via the associated martingale problem. For this the above mentioned strong convergence of generators is essential.

Staggered Ladder Spectra

We exactly solve a Fokker-Planck equation by determining its eigenvalues and eigenfunctions: we construct nonlinear second-order differential operators which act as raising and lowering operators, generating ladder spectra for the odd- and even-parity states. The ladders are staggered: the odd-even separation differs from even-odd. The Fokker-Planck equation corresponds, in the limit of weak damping, to a generalized Ornstein-Uhlenbeck process where the random force depends upon position as well as time. The process describes damped stochastic acceleration, and exhibits anomalous diffusion at short times and a stationary non-Maxwellian momentum distribution.

Equilibrium Fluctuations for a System of Harmonic Oscillators with Conservative Noise

We investigate the harmonic chain forced by a multiplicative noise, the evolution is given by an infinite system of stochastic differential equations. Total energy and deformation are preserved, the conservation of momentum is destroyed by the noise. Gaussian product measures are the extremal stationary states of this model. Equilibrium fluctuations of the conserved fields at a diffusive scaling are described by a couple of generalized Ornstein-Uhlenbeck processes.

Branching Processes with Immigration and Related Topics

This is a survey on the recent progresses in the study of branching processes with immigration, generalized Ornstein-Uhlenbeck processes, and affine Markov processes. We mainly focus on the applications of skew convolution semigroups and the connections in those processes.

Lévy integrals and the stationarity of generalised Ornstein–Uhlenbeck processes

The generalised Ornstein–Uhlenbeck process constructed from a bivariate Lévy process ( ξ t , η t ) t ⩾ 0 is defined as V t = e - ξ t ∫ 0 t e ξ s - d η s + V 0 , t ⩾ 0 , where V 0 is an independent starting random variable. The stationarity of the process is closely related to the convergence or divergence of the Lévy integral ∫ 0 ∞ e - ξ t - d η t . We make precise this relation in the general case, showing that the conditions are not in general equivalent, though they are for example if ξ and η are independent. Characterisations are expressed in terms of the Lévy measure of ( ξ , η ) . Conditions for the moments of the strictly stationary distribution to be finite are given, and the autocovariance function and the heavy-tailed behaviour of the stationary solution are also studied.

On a martingale associated to generalized Ornstein–Uhlenbeck processes and an application to finance

In this paper we study the two-dimensional joint distribution of the first passage time of a constant level by spectrally negative generalized Ornstein–Uhlenbeck processes and their primitive stopped at this first passage time. By using martingales techniques, we show an explicit expression of the Laplace transform of the distribution in terms of new special functions. Finally, we give an application in finance which consists of computing the Laplace transform of the price of an European call option on the maximum on the yield in the generalized Vasicek model. The stable case is studied in more detail.

Generalized Mehler Semigroups and Catalytic Branching Processes with Immigration

Skew convolution semigroups play an important role in the study of generalized Mehler semigroups and Ornstein–Uhlenbeck processes. We give a characterization for a general skew convolution semigroup on a real separable Hilbert space whose characteristic functional is not necessarily differentiable at the initial time. A connection between this subject and catalytic branching superprocesses is established through fluctuation limits, providing a rich class of non-differentiable skew convolution semigroups. Path regularity of the corresponding generalized Ornstein–Uhlenbeck processes in different topologies is also discussed.

Complete description of a generalized Ornstein?Uhlenbeck process related to the nonextensive Gaussian entropy

We consider a generalized Ornstein–Uhlenbeck process described by a nonlinear Fokker–Planck equation related to the one-parametric, nonextensive Gaussian entropy, which is a special case of the two-parametric Sharma–Mittal entropy. We derive the entire hierarchy of distribution functions for that process and, in doing so, derive a complete description of a stochastic process related to a nonextensive entropy measure.

Complete description of a generalized Ornstein–Uhlenbeck process related to the nonextensive Gaussian entropy

We consider a generalized Ornstein–Uhlenbeck process described by a nonlinear Fokker–Planck equation related to the one-parametric, nonextensive Gaussian entropy, which is a special case of the two-parametric Sharma–Mittal entropy. We derive the entire hierarchy of distribution functions for that process and, in doing so, derive a complete description of a stochastic process related to a nonextensive entropy measure.

Stochastic Perturbations of Line Solitons of KP

We consider the properties of localized solutions of the KP equation coupled to a stochastic noise. Corresponding to white noise, we find that the traveling waves are destroyed asymptotically, and we determine the distribution of the wave position and the arrival time. For generalized Ornstein–Uhlenbeck processes, we show that the only effect of noise is to render the asymptotic position random; in particular, when the noise has a sufficiently strong attenuation mechanism, the random wave coincides asymptotically with the unperturbed one. We also consider linearization of the corresponding Cauchy problem in the plane corresponding to this kind of initial data.

Equilibrium fluctuations for lattice gases

The authors in a previous paper proved the hydrodynamic incompressible limit in d⩾3 for a thermal lattice gas, namely a law of large numbers for the density, velocity fields and energy. In this paper the equilibrium fluctuations for this model are studied and a central limit theorem is proved for a suitable modification of the vector fluctuation field ζ ε ( t), whose components are the density, velocity and energy fluctuations fields. We consider a modified fluctuation field ξ ε ( t)=exp{− ε −1 tE} ζ ε , where E is the linearized Euler operator around the equilibrium, and prove that ξ ε ( t) converges to a vector generalized Ornstein–Uhlenbeck process ξ( t), which is formally solution of the stochastic differential equation dξ(t)=Nξ(t) dt+B dW t , with BB ∗=−2NC , where C is the compressibility matrix, N is a matrix whose entries are second order differential operators and W t is a mean zero Gaussian field. The relation −2NC=BB ∗ is the fluctuation-dissipation relation.

Generalized Ornstein Uhlenbeck processes and associated self-similar processes

We consider three types of generalized Ornstein–Uhlenbeck processes: the stationary process obtained from the Lamperti transformation of fractional Brownian motion, the process with stretched exponential covariance and the process obtained from the solution of the fractional Langevin equation. These stationary Gaussian processes have many common properties, such as the fact that their local covariances share a similar structure and they exhibit identical spectral densities at large frequency limit. In addition, the generalized Ornstein–Uhlenbeck processes can be shown to be local stationary representations of fractional Brownian motion. Two new self-similar Gaussian processes, in addition to fractional Brownian motion, are obtained by applying the (inverse) Lamperti transformation to the generalized Ornstein–Uhlenbeck processes. We study some of the properties of these self-similar processes such as the long-range dependence. We give a simulation of their sample paths based on numerical Karhunan–Loeve expansion.

Modeling electricity loads in California: a continuous-time approach

In this paper we address the issue of modeling electricity loads and prices with diffusion processes. More specifically, we study models which belong to the class of generalized Ornstein–Uhlenbeck processes. After comparing properties of simulated paths with those of deseasonalized data from the California power market and performing out-of-sample forecasts we conclude that, despite certain advantages, the analyzed continuous-time processes are not adequate models of electricity load and price dynamics.

Equilibrium fluctuations of asymmetric simple exclusion processes in dimension d≥3

We consider an asymmetric exclusion process in dimension d ≥ 3 under diffu- sive rescaling starting from the Bernoulli product measure with density 0 <α <1. We prove that the density fluctuation field Y N t converges to a generalized Ornstein-Uhlenbeck process, which is formally the solution of the stochastic differential equation dYt = AYt dt + dB ∇ , where A is a second order differential operator and B ∇ t is a mean zero Gaussian field with known covariances.

A Generalized Ornstein-Uhlenbeck Process of Yield Rates Calibrated with Strips

In this article, we present a Gaussian multivariate factor model of the term structure of interest rates and we present an explicit formula for a state-price probability density function used in no-arbitrage pricing. This leads to an explicit expression for Green's function. The model's linear and Gaussian structure yields a simple model with formulas for no-arbitrage pricing that are readily implemented. Moreover, estimation and calibration is relatively easy to do. Using yield data on stripped bonds, we calibrate our model and demonstrate that the Gaussian assumption is not unreasonable. The model accurately explains the dynamics of the whole yield curve for periods of at least one year.