Generalized Ornstein Uhlenbeck Research Articles

On moments of integrals with respect to Markov additive processes and of Markov modulated generalized Ornstein–Uhlenbeck processes

We establish sufficient conditions for the existence, and derive explicit formulas for the κ’th moments, κ≥1, of Markov modulated generalized Ornstein–Uhlenbeck processes as well as their stationary distributions. In particular, the running mean, the autocovariance function, and integer moments of the stationary distribution are derived in terms of the characteristics of the driving Markov additive process.Our derivations rely on new general results on moments of Markov additive processes and (multidimensional) integrals with respect to Markov additive processes.

Gaussian Fluctuations for Interacting Particle Systems with Singular Kernels

We consider the asymptotic behaviour of the fluctuations for the empirical measures of interacting particle systems with singular kernels. We prove that the sequence of fluctuation processes converges in distribution to a generalized Ornstein–Uhlenbeck process. Our result considerably extends classical results to singular kernels, including the Biot–Savart law. The result applies to the point vortex model approximating the 2D incompressible Navier–Stokes equation and the 2D Euler equation. We also obtain Gaussianity and optimal regularity of the limiting Ornstein–Uhlenbeck process. The method relies on the martingale approach and the Donsker–Varadhan variational formula, which transfers the uniform estimate to some exponential integrals. Estimation of those exponential integrals follows by cancellations and combinatorics techniques and is of the type of the large deviation principle.

Scaling limits and fluctuations of a family of N-urn branching processes

In this paper, we are concerned with a family of N-urn branching processes, where some particles are initially placed in N urns, and then each particle gives birth to several new particles in some urn when dies. This model includes the N-urn Ehrenfest model and N-urn branching random walk as special cases. We show that the scaling limit of the process is driven by a C(T)-valued linear ordinary differential equation and the fluctuation of the process is driven by a generalized Ornstein–Uhlenbeck process in the dual of C∞(T), where T=(0,1] is the one-dimensional torus. A crucial step for the proofs of the above main results is to show that numbers of particles in different urns are approximately independent. As applications of our main results, the limit theorems of the hitting times of the process are also discussed.

Magnetospheric Multiscale Observations of Markov Turbulence on Kinetic Scales

In our previous studies we have examined solar wind and magnetospheric plasma turbulence, including Markovian character on large inertial magnetohydrodynamic scales. Here we present the results of the statistical analysis of magnetic field fluctuations in the Earth’s magnetosheath, based on the Magnetospheric Multiscale mission at much smaller kinetic scales. Following our results on spectral analysis with very large slopes of about −16/3, we apply a Markov-process approach to turbulence in this kinetic regime. It is shown that the Chapman–Kolmogorov equation is satisfied and that the lowest-order Kramers–Moyal coefficients describing drift and diffusion with a power-law dependence are consistent with a generalized Ornstein–Uhlenbeck process. The solutions of the Fokker–Planck equation agree with experimental probability density functions, which exhibit a universal global scale invariance through the kinetic domain. In particular, for moderate scales we have the kappa distribution described by various peaked shapes with heavy tails, which, with large values of the kappa parameter, are reduced to the Gaussian distribution for large inertial scales. This shows that the turbulence cascade can be described by the Markov processes also on very small scales. The obtained results on kinetic scales may be useful for a better understanding of the physical mechanisms governing turbulence.

Generalized Ornstein–Uhlenbeck semigroups in weighted Lp-spaces on Riemannian manifolds

Let E be a Hermitian vector bundle over a Riemannian manifold M with metric g, and let ∇ be a metric covariant derivative on E. We study the generalized Ornstein–Uhlenbeck differential expression P∇=∇†∇u+∇(dϕ)♯u−∇Xu+Vu, where ∇† is the formal adjoint of ∇, (dϕ)♯ is the vector field corresponding to dϕ via g, X is a smooth real vector field on M, and V is a self-adjoint locally integrable section of the endomorphsim bundle EndE. In the setting of a geodesically complete M, we establish a sufficient condition for the equality of the maximal and minimal realizations of P∇ in the (weighted) space ΓLμp(E) of Lμp-type sections of E, where 1<p<∞ and dμ=e−ϕdνg, with νg being the usual volume measure. Furthermore, we show that (the negative of) the maximal realization −Hp,max generates an analytic quasi-contractive semigroup in ΓLμp(E), 1<p<∞. Additionally, in the same context, we establish a coercivity estimate, which leads to the so-called separation property of the covariant Schrödinger operator (that is, P∇ with X≡0 and ϕ≡0) in the (unweighted) space ΓLp(E), 1<p<∞. With the generation result at our disposal, we describe a Feynman–Kac representation for the Lμp-semigroup generated by Hp,max, 1<p<∞. For the Ornstein–Uhlenbeck differential expression acting on functions, that is, Pd=Δu+(dϕ)♯u−Xu+Vu, where Δ is the (non-negative) scalar Laplacian on M and V is a locally integrable real-valued function, we consider another way of realizing Pd as an operator in Lμp(M) and, by imposing certain geometric conditions on M, we prove another semigroup generation result. The study of the mentioned realization of Pd depends, among other things, on the fulfillment of the so-called Lp-Calderón–Zygmund inequality on M.

On the maximal operator of a general Ornstein–Uhlenbeck semigroup

AbstractIf Q is a real, symmetric and positive definite $$n\times n$$ n × n matrix, and B a real $$n\times n$$ n × n matrix whose eigenvalues have negative real parts, we consider the Ornstein–Uhlenbeck semigroup on $$\mathbb {R}^n$$ R n with covariance Q and drift matrix B. Our main result says that the associated maximal operator is of weak type (1, 1) with respect to the invariant measure. The proof has a geometric gist and hinges on the “forbidden zones method” previously introduced by the third author.

Cutoff Thermalization for Ornstein\u2013Uhlenbeck Systems with Small L\xe9vy Noise in the Wasserstein Distance

This article establishes cutoff thermalization (also known as the cutoff phenomenon) for a class of generalized Ornstein–Uhlenbeck systems (X^varepsilon _t(x))_{tgeqslant 0} with varepsilon -small additive Lévy noise and initial value x. The driving noise processes include Brownian motion, alpha -stable Lévy flights, finite intensity compound Poisson processes, and red noises, and may be highly degenerate. Window cutoff thermalization is shown under mild generic assumptions; that is, we see an asymptotically sharp infty /0-collapse of the renormalized Wasserstein distance from the current state to the equilibrium measure mu ^varepsilon along a time window centered on a precise varepsilon -dependent time scale mathfrak {t}_varepsilon . In many interesting situations such as reversible (Lévy) diffusions it is possible to prove the existence of an explicit, universal, deterministic cutoff thermalization profile. That is, for generic initial data x we obtain the stronger result mathcal {W}_p(X^varepsilon _{t_varepsilon + r}(x), mu ^varepsilon ) cdot varepsilon ^{-1} rightarrow Kcdot e^{-q r} for any rin mathbb {R} as varepsilon rightarrow 0 for some spectral constants K, q>0 and any pgeqslant 1 whenever the distance is finite. The existence of this limit is characterized by the absence of non-normal growth patterns in terms of an orthogonality condition on a computable family of generalized eigenvectors of mathcal {Q}. Precise error bounds are given. Using these results, this article provides a complete discussion of the cutoff phenomenon for the classical linear oscillator with friction subject to varepsilon -small Brownian motion or alpha -stable Lévy flights. Furthermore, we cover the highly degenerate case of a linear chain of oscillators in a generalized heat bath at low temperature.

Continuity properties and the support of killed exponential functionals

For two independent Lévy processes ξ and η and an exponentially distributed random variable τ with parameter q>0, independent of ξ and η, the killed exponential functional is given by Vq,ξ,η≔∫0τe−ξs−dηs. Interpreting the case q=0 as τ=∞, the random variable Vq,ξ,η is a natural generalisation of the exponential functional ∫0∞e−ξs−dηs, the law of which is well-studied in the literature as it is the stationary distribution of a generalised Ornstein–Uhlenbeck process. In this paper we show that also the law of the killed exponential functional Vq,ξ,η arises as a stationary distribution of a solution to a stochastic differential equation, thus establishing a close connection to generalised Ornstein–Uhlenbeck processes. Moreover, the support and continuity of the law of killed exponential functionals is characterised, and many sufficient conditions for absolute continuity are derived. We also obtain various new sufficient conditions for absolute continuity of ∫0te−ξs−dηs for fixed t≥0, as well as for integrals of the form ∫0∞f(s)dηs for deterministic functions f. Furthermore, applying the same techniques to the case q=0, new results on the absolute continuity of the improper integral ∫0∞e−ξs−dηs are derived.

Generalized Ornstein–Uhlenbeck operators perturbed by multipolar inverse square potentials in $$L^{2}$$-spaces

Generalized Ornstein–Uhlenbeck operators perturbed by multipolar inverse square potentials in $$L^{2}$$-spaces

Sensitivity analysis in the infinite dimensional Heston model

We consider the infinite dimensional Heston stochastic volatility model proposed in Ref. 7. The price of a forward contract on a non-storable commodity is modeled by a generalized Ornstein–Uhlenbeck process in the Filipović space with this volatility. We prove a representation formula for the forward price. Then we consider prices of options written on these forward contracts and we study sensitivity analysis with computation of the Greeks with respect to different parameters in the model. Since these parameters are infinite dimensional, we need to reinterpret the meaning of the Greeks. For this we use infinite dimensional Malliavin calculus and a randomization technique.

Efficient Bayesian inference of general Gaussian models on large phylogenetic trees

Phylogenetic comparative methods correct for shared evolutionary history among a set of nonindependent organisms by modeling sample traits as arising from a diffusion process along the branches of a possibly unknown history. To incorporate such uncertainty, we present a scalable Bayesian inference framework under a general Gaussian trait evolution model that exploits Hamiltonian Monte Carlo (HMC). HMC enables efficient sampling of the constrained model parameters and takes advantage of the tree structure for fast likelihood and gradient computations, yielding algorithmic complexity linear in the number of observations. This approach encompasses a wide family of stochastic processes, including the general Ornstein–Uhlenbeck (OU) process, with possible missing data and measurement errors. We implement inference tools for a biologically relevant subset of all these models into the BEAST phylogenetic software package and develop model comparison through marginal likelihood estimation. We apply our approach to study the morphological evolution in the superfamily of Musteloidea (including weasels and allies) as well as the heritability of HIV virulence. This second problem furnishes a new measure of evolutionary heritability that demonstrates its utility through a targeted simulation study.

Hydrodynamics of the weakly asymmetric normalized binary contact path process

We are concerned with the weakly asymmetric normalized binary contact path process. The symmetric version was introduced in Griffeath (1983). The model describes the spread of an infectious disease on the lattice Zd. The configuration at each site x∈Zd takes value in [0,∞). Site x is recovered at rate 1, and is infected by its neighbor x±ei at rate λ∓λ1∕N, where {ei}1≤i≤d is the canonical basis of the d-dimensional lattice, λ,λ1 are non-negative constants and N is the scaling parameter. When the infection occurs, the seriousness of the disease at site x is added with that of y. When there is neither recovery nor infection occurring during some time interval, the value at site x evolves according to some ODE. We prove that for d≥3 and large value λ, the empirical measure of the process, under diffusive scaling, converges in probability to a deterministic measure whose density is the unique weak solution to a linear parabolic equation, while for small λ and in all dimensions, the limit is zero. We also conjecture that the fluctuations field in the former case is driven by a generalized Ornstein–Uhlenbeck process, while a rigorous proof is absent. The main difficulty in proving the hydrodynamics is to prove the absolute continuity of the limiting path, where the theory of linear systems introduced in Liggett (1985) is utilized.

Weak limits of random coefficient autoregressive processes and their application in ruin theory

We prove that a large class of discrete-time insurance surplus processes converge weakly to a generalized Ornstein–Uhlenbeck process, under a suitable re-normalization and when the time-step goes to 0. Motivated by ruin theory, we use this result to obtain approximations for the moments, the ultimate ruin probability and the discounted penalty function of the discrete-time process.

Ruin probabilities for a Lévy-driven generalised Ornstein–Uhlenbeck process

We study the asymptotics of the ruin probability for a process which is the solution of a linear SDE defined by a pair of independent Levy processes. Our main interest is a model describing the evolution of the capital reserve of an insurance company selling annuities and investing in a risky asset. Let $\beta >0$ be the root of the cumulant-generating function $H$ of the increment $V_{1}$ of the log-price process. We show that the ruin probability admits the exact asymptotic $Cu^{-\beta }$ as the initial capital $u\to \infty $, assuming only that the law of $V_{T}$ is non-arithmetic without any further assumptions on the price process.

Inference in a multivariate generalized mean-reverting process with a change-point

In this paper, we study inference problem about the drift parameter matrix in multivariate generalized Ornstein–Uhlenbeck processes with an unknown change-point. In particular, we consider the case where the parameter matrix may satisfy some restrictions. Thus, we generalize in five ways some recent findings about univariate generalized Ornstein–Uhlenbeck processes. First, the target parameter is a matrix and we derive a sufficient condition for the existence of the unrestricted estimator (UE) and the restricted estimator (RE). Second, we establish the joint asymptotic normality of the UE and the RE under a collection of local alternatives. Third, we construct a test for testing the uncertain restriction. The proposed test is also useful for testing the absence of the change-point. Fourth, we derive the asymptotic power of the proposed test and we prove that it is consistent. Fifth, we propose the shrinkage estimators (SEs) and we prove that SEs dominate the UE. Finally, we conduct some simulation studies which corroborate our theoretical findings.

An $$L^p$$ L p -Theory for Generalized Ornstein–Uhlenbeck Operators with Nonnegative Singular Potentials

This paper is concerned with some properties of the generalized Ornstein–Uhlenbeck operator $$\begin{aligned} -A_{\Phi ,G}+\nu V:=-\Delta +\nabla \Phi \cdot \nabla -G\cdot \nabla +\nu V, \end{aligned}$$ with nonnegative singular potential $$\nu V(x)$$ in the weighted space $$L^{p}({\mathbb {R}}^{N},\mu )$$ , $$1<p<\infty $$ , where $$\mu (dx)=e^{-\Phi (x)}dx$$ . Sufficient conditions ensuring the m-accretivity, m-sectoriality and m-dispersivity of $$-A_{\Phi ,G}+\nu V$$ in $$L^{p}({\mathbb {R}}^{N},\mu )$$ are presented. Particularly, it is shown that $$A_{\Phi ,G}-\nu V$$ with a suitable domain is the generator of a quasi-contractive and positivity preserving analytic $$C_0$$ -semigroup in $$L^{p}({\mathbb {R}}^{N},\mu )$$ . Further, generation of quasi-contractive analytic semigroups by $$A_{\Phi ,G}-(\nu +k)V$$ in $$L^{p}({\mathbb {R}}^{N},\mu )$$ , for some $$\nu >0$$ and $$ k\in {\mathbb {C}}$$ , is proven. The results improve and complete the recent results established in Metafune et al. (Adv Differ Equ 10(10):1131–1164, 2005) when $$V\equiv 0$$ and Kojima and Yokota (J Math Anal Appl 364(2):618–629, 2010) and Sobajima and Yokota (J Math Anal Appl 403(2):606–618, 2013) when $$0\le V\in C^1({\mathbb {R}}^N)$$ . Some examples where our results can be applied are provided, including Kolmogorov operators and operators with polynomially growing drift term.

Ergodic aspects of some Ornstein–Uhlenbeck type processes related to Lévy processes

This work concerns the Ornstein–Uhlenbeck type process associated to a positive self-similar Markov process (X(t))t≥0 which drifts to ∞, namely U(t)≔e−tX(et−1). We point out that U is always a (topologically) recurrent ergodic Markov process. We identify its invariant measure in terms of the law of the exponential functional Iˆ≔∫0∞exp(ξˆs)ds, where ξˆ is the dual of the real-valued Lévy process ξ related to X by the Lamperti transformation. This invariant measure is infinite (i.e. U is null-recurrent) if and only if ξ1∉L1(P). In that case, we determine the family of Lévy processes ξ for which U fulfills the conclusions of the Darling–Kac theorem. Our approach relies crucially on a remarkable connection due to Patie (Patie, 2008) with another generalized Ornstein–Uhlenbeck process that can be associated to the Lévy process ξ, and properties of time-substitutions based on additive functionals.

An Elliptic PDE with Convex Solutions

AbstractUsing a mixture of classical and probabilistic techniques, we investigate the convexity of solutions to the elliptic partial differential equation associated with a certain generalized Ornstein–Uhlenbeck process.

Stochastic Navier–Stokes equations with Caputo derivative driven by fractional noises

In this paper, we consider the extended stochastic Navier–Stokes equations with Caputo derivative driven by fractional Brownian motion. We firstly derive the pathwise spatial and temporal regularity of the generalized Ornstein–Uhlenbeck process. Then we discuss the existence, uniqueness, and Hölder regularity of mild solutions to the given problem under certain sufficient conditions, which depend on the fractional order α and Hurst parameter H. The results obtained in this study improve some results in existing literature.

The asymptotic smile of a multiscaling stochastic volatility model

We consider a stochastic volatility model which captures relevant stylized facts of financial series, including the multi-scaling of moments. The volatility evolves according to a generalized Ornstein–Uhlenbeck processes with super-linear mean reversion.Using large deviations techniques, we determine the asymptotic shape of the implied volatility surface in any regime of small maturity t→0 or extreme log-strike |κ|→∞ (with bounded maturity). Even if the price has continuous paths, out-of-the-money implied volatility diverges for small maturity, producing a very pronounced smile.