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.     

How to test that a given process is an Ornstein–Uhlenbeck process

Statistical Inference for Stochastic Processes - We show asymptotic distributions of the residual process in Ornstein–Uhlenbeck model, when the model is true. These distributions are of...     

Parameter estimation of Ornstein–Uhlenbeck process generating a stochastic graph

Given Y a graph process defined by an incomplete information observation of a multivariate Ornstein–Uhlenbeck process X, we investigate whether we can estimate the parameters of X. We define two statistics of Y. We prove convergence properties and show how these can be used for parameter inference. Finally, numerical tests illustrate our results and indicate possible extensions and applications.     

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.     

Parameter identification for the Hermite Ornstein–Uhlenbeck process

Statistical Inference for Stochastic Processes - By using the analysis on Wiener chaos, we study the behavior of the quadratic variations of the Hermite Ornstein–Uhlenbeck process, which is...     

Sharp large deviations for the non-stationary Ornstein–Uhlenbeck process

For the Ornstein–Uhlenbeck process, the asymptotic behavior of the maximum likelihood estimator of the drift parameter is totally different in the stable, unstable, and explosive cases. Notwithstanding this trichotomy, we investigate sharp large deviation principles for this estimator in the three situations. In the explosive case, we exhibit a very unusual rate function with a shaped flat valley and an abrupt discontinuity point at its minimum.     

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.     

Hypercontractivity for a quantum Ornstein–Uhlenbeck semigroup

We prove hypercontractivity for a quantum Ornstein–Uhlenbeck semigroup on the entire algebra of bounded operators on a separable Hilbert space h. We exploit the particular structure of the spectrum together with hypercontractivity of the corresponding birth and death process and a proper decomposition of the domain. Then we deduce a logarithmic Sobolev inequality for the semigroup and gain an elementary estimate of the best constant.     

Optimal dividends and bankruptcy procedures: Analysis of the Ornstein–Uhlenbeck process

This paper investigates the impact of bankruptcy procedures on optimal dividend barrier policies. We specifically focus on Chapter 11 of the US Bankruptcy Code, which allows a firm in default to continue its business for a certain period of time. Our model is based on the surplus of a firm that earns investment income at a constant rate of credit interest when it is in a creditworthy condition. The firm pays a debit interest rate that depends on the deficit level when it is in financial distress. Thus, the surplus follows an Ornstein–Uhlenbeck (OU) process with a negative surplus-dependent mean-reverting rate. Default and liquidation are modeled as distinguishable events by using an excursion time or occupation time framework. This paper demonstrates how the optimal dividend barrier can be obtained by deriving a closed-form solution for the dividend value function. It also characterizes the distributional property and expectation of bankruptcy time subject to the bankruptcy procedure. Our numerical examples show that under an optimal dividend barrier strategy, the bankruptcy procedure may not prolong the expected bankruptcy time in some situations.     

Ornstein–Uhlenbeck Path Integral and Its Application

Introducing a path integral for the Ornstein–Uhlenbeck process distorted by a potential V(x), we find out the T limit of the probability distributions of X[]:=1/T ₀ ^T V((t))dt for Ornstein–Uhlenbeck process (t), with appropriate values of the exponent that depend on V. The results are compared with those for the Wiener process.     

Bayesian inference of the fractional Ornstein–Uhlenbeck process under a flow sampling scheme

Using recent developments in econometrics and computational statistics we consider the estimation of the fractional Ornstein–Uhlenbeck process under a flow sampling scheme. To address the problem, we adopt throughout the paper an exact discretization approach. A flow sampling scheme arises, for example, naturally in modelling asset prices in continuous time since the time integral over successive observations defines the observable increments of asset log-prices. Exact discretization delivers an ARIMA(1,1,1) model for log-prices with a fractional driving noise. Building on the resulting exact discretization formulae and covariance function, a new Markov Chain Monte Carlo scheme is proposed and apply it to investigate the properties of both the time and frequency domain likelihoods/posteriors. For the exact discrete model, we adopt a general sampling interval of length h. This allows us to determine the optimal choice of h independent of the sample size. To illustrate the methods, with no ambition to a comprehensive data analysis, we use high frequency stock price data showing the relevance of aggregation over time issues in modelling asset prices.     

Exact asymptotic bias for estimators of the Ornstein–Uhlenbeck process

We study the bias and the bias derivative for a family \({\mathcal{F}}\) of asymptotically efficient estimators of the Ornstein–Uhlenbeck process. That family contains the maximum likelihood, the conditional maximum likelihood and the empirical estimators. We show that, if g(θ _T ) is an estimator of g(θ), where θ is the parameter and \({\theta_{T} \in \mathcal{F}}\), then, under mild conditions,

$T\,E\left[g(\theta_{T})-g(\theta)\right]\xrightarrow[T\rightarrow\infty]{}c_{\theta}g^{\prime}(\theta)+\theta{g}^{\prime\prime}(\theta),$

where c _θ is an explicit constant that only depends on the choice of θ _T . In particular, if θ _T is one of the three previous estimators, one has

$T\,E_{\theta}(\theta_{T}-\theta)\xrightarrow[T\rightarrow\infty]\,2.$

     

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.     

Phylogenetic Ornstein–Uhlenbeck regression curves

Regression curves for studying trait relationships are developed herein. The adaptive evolution model is considered as an Ornstein–Uhlenbeck system whose parameters are estimated by a novel engagement of generalized least-squares and optimization. Our algorithm is implemented to ecological data.     

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.