Generalized Ornstein-Uhlenbeck Process 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.

Drift estimation for a class of generalized Ornstein-Uhlenbeck process with fluctuating exponential trend

Drift estimation for Generalized Ornstein-Uhlenbeck processes with Lévy noise and up to 3rd-order autoregressive behavior are studied. Under exponential trend and cosine oscillations, we propose two discrete-time versions of the MLE for the drift parameter. Both estimates are derived using jump filtering techniques. Simulation results showing their performance are included.

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.

Ornstein-Uhlenbeck process and generalizations: Particle dynamics under comb constraints and stochastic resetting.

The Ornstein-Uhlenbeck process is interpreted as Brownian motion in a harmonic potential. This Gaussian Markov process has a bounded variance and admits a stationary probability distribution, in contrast to the standard Brownian motion. It also tends to a drift towards its mean function, and such a process is called mean reverting. Two examples of the generalized Ornstein-Uhlenbeck process are considered. In the first one, we study the Ornstein-Uhlenbeck process on a comb model, as an example of the harmonically bounded random motion in the topologically constrained geometry. The main dynamical characteristics (as the first and the second moments) and the probability density function are studied in the framework of both the Langevin stochastic equationand the Fokker-Planck equation. The second example is devoted to the study of the effects of stochastic resetting on the Ornstein-Uhlenbeck process, including stochastic resetting in the comb geometry. Here the nonequilibrium stationary state is the main question in task, where the two divergent forces, namely, the resetting and the drift towards the mean, lead to compelling results in the cases of both the Ornstein-Uhlenbeck process with resetting and its generalization on the two-dimensional comb structure.

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.

Markov-modulated generalized Ornstein-Uhlenbeck processes and an application in risk theory

By embedding a Markov-modulated random recurrence equation in continuous time, we derive the Markov-modulated generalized Ornstein-Uhlenbeck process. This process turns out to be the unique solution of a stochastic differential equation driven by a bivariate Markov-additive process. We present this stochastic differential equation as well as its solution explicitely in terms of the driving Markov-additive process. Moreover, we give necessary and sufficient conditions for strict stationarity of the Markov-modulated generalized Ornstein-Uhlenbeck process, and prove that its stationary distribution is given by the distribution of a certain exponential functional of Markov-additive processes. Finally, we propose a Markov-modulated risk model with investment that generalizes Paulsen’s risk process to a Markov-switching environment, and derive a formula for the ruin probability in this risk model.

Optimal liquidation of foreign currencies when FX rates follow a generalised Ornstein-Uhlenbeck process

Optimal liquidation of foreign currencies when FX rates follow a generalised Ornstein-Uhlenbeck process

Estimation and Testing in Multivariate Generalized Ornstein-Uhlenbeck Processes with Change-Points

Estimation and Testing in Multivariate Generalized Ornstein-Uhlenbeck Processes with Change-Points

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.

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.

Higher order fluctuation fields and orthogonal duality polynomials

Inspired by the works in [2] and [11] we introduce what we call k-th-order fluctuation fields and study their scaling limits. This construction is done in the context of particle systems with the property of orthogonal self-duality. This type of duality provides us with a setting in which we are able to interpret these fields as some type of discrete analogue of powers of the well-known density fluctuation field. We show that the weak limit of the k-th order field satisfies a recursive martingale problem that corresponds to the SPDE associated with the kth-power of a generalized Ornstein-Uhlenbeck process.

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.

Exact Simulation of Variance Gamma-Related OU Processes: Application to the Pricing of Energy Derivatives

ABSTRACT In this study we use a three-step procedure that relates the self-decomposability of the stationary law of a generalized Ornstein-Uhlenbeck process to the transition law of such processes. Based on this procedure and the results of Qu, Dassios, and Zhao (2019), we derive the exact simulation, without numerical inversion, of the skeleton of a Variance Gamma and of a symmetric Variance Gamma driven Ornstein-Uhlenbeck process. Extensive numerical experiments are reported to demonstrate the accuracy and efficiency of our algorithms. These results are instrumental to simulate the spot price dynamics in energy markets and to price Asian options and gas storages by Monte Carlo simulations in a framework similar to the one discussed in Cummins, Kiely and Murphy (2017, 2018).

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.

A result on power moments of Lévy-type perpetuities and its application to the Lp -convergence of Biggins’ martingales in branching Lévy processes

L\'evy-type perpetuities being the a.s. limits of particular generalized Ornstein-Uhlenbeck processes are a natural continuous-time generalization of discrete-time perpetuities. These are random variables of the form $S:=\int_{[0,\infty)}e^{-X_{s-}}{\mathrm{d}}Z_s$, where $(X,Z)$ is a two-dimensional L\'evy process, and $Z$ is a drift-free L\'evy process of bounded variation. We prove an ultimate criterion for the finiteness of power moments of $S$. This result and the previously known assertion due to Erickson and Maller (2005) concerning the a.s. finiteness of $S$ are then used to derive ultimate necessary and sufficient conditions for the $L_p$-convergence for $p>1$ and $p=1$, respectively, of Biggins' martingales associated to branching L\'evy processes. In particular, we provide final versions of results obtained recently by Bertoin and Mallein (2018).

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.