Comparison of the LS-based estimators and the MLE for the fractional Ornstein–Uhlenbeck process

Statistical Inference for Stochastic Processes - We deal with the fractional Ornstein–Uhlenbeck (fO–U) process driven by the fractional Brownian motion (fBm), where the drift parameter...     

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.     

Distributions of the maximum likelihood and minimum contrast estimators associated with the fractional Ornstein–Uhlenbeck process

We discuss some inference problems associated with the fractional Ornstein–Uhlenbeck (fO–U) process driven by the fractional Brownian motion (fBm). In particular, we are concerned with the estimation of the drift parameter, assuming that the Hurst parameter $H$ is known and is in $[1/2, 1)$ . Under this setting we compute the distributions of the maximum likelihood estimator (MLE) and the minimum contrast estimator (MCE) for the drift parameter, and explore their distributional properties by paying attention to the influence of $H$ and the sampling span $M$ . We also deal with the ordinary least squares estimator (OLSE) and examine the asymptotic relative efficiency. It is shown that the MCE is asymptotically efficient, while the OLSE is inefficient. We also consider the unit root testing problem in the fO–U process and compute the power of the tests based on the MLE and MCE.     

Ornstein–Uhlenbeck process,Cauchy process,and Ornstein–Uhlenbeck–Cauchy process on a circle

By a simple mathematical method, we obtain the transition probability density functions of the Ornstein–Uhlenbeck process, Cauchy process, and Ornstein–Uhlenbeck–Cauchy process on a circle.     

Local asymptotic mixed normality for discretely observed non-recurrent Ornstein–Uhlenbeck processes

Consider non-recurrent Ornstein–Uhlenbeck processes with unknown drift and diffusion parameters. Our purpose is to estimate the parameters jointly from discrete observations with a certain asymptotics. We show that the likelihood ratio of the discrete samples has the uniform LAMN property, and that some kind of approximated MLE is asymptotically optimal in a sense of asymptotic maximum concentration probability. The estimator is also asymptotically efficient in ergodic cases.     

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.     

Functional calculus for some perturbations of the Ornstein–Uhlenbeck operator

In a recent paper García-Cuerva et al. have shown that for every p in (1,∞) the symmetric finite-dimensional Ornstein–Uhlenbeck operator has a bounded holomorphic functional calculus on L ^p in the sector of angle . We prove a similar result for some perturbations of the Ornstein–Uhlenbeck operator. Work partially supported by the Progetto Cofinanziato MIUR “Analisi Armonica” and the Gruppo Nazionale INdAM per l’Analisi Matematica, la Probabilitàe le loro Applicazioni.     

Comparison of Spaces of Hardy Type for the Ornstein–Uhlenbeck Operator

Denote by γ the Gauss measure on ℝ ⁿ and by ${\mathcal{L}}${\mathcal{L}} the Ornstein–Uhlenbeck operator. In this paper we introduce a Hardy space \mathfrakh¹g{{\mathfrak{h}}^1}{{\rm \gamma}} of Goldberg type and show that for each u in ℝ ∖ {0} and r > 0 the operator (rI+L)^iu(r{\mathcal{I}}+{\mathcal{L}})^{iu} is unbounded from \mathfrakh¹g{{\mathfrak{h}}^1}{{\rm \gamma}} to L ¹γ. This result is in sharp contrast both with the fact that (rI+L)^iu(r{\mathcal{I}}+{\mathcal{L}})^{iu} is bounded from H ¹γ to L ¹γ, where H ¹γ denotes the Hardy type space introduced in Mauceri and Meda (J Funct Anal 252:278–313, 2007), and with the fact that in the Euclidean case (rI-D)^iu(r{\mathcal{I}}-\Delta)^{iu} is bounded from the Goldberg space \mathfrakh¹\mathbbRⁿ{{\mathfrak{h}}^1}{{\mathbb{R}}^n} to L ¹ℝ ⁿ . We consider also the case of Riemannian manifolds M with Riemannian measure μ. We prove that, under certain geometric assumptions on M, an operator T{\mathcal{T}}, bounded on L ² μ, and with a kernel satisfying certain analytic assumptions, is bounded from H ¹ μ to L ¹ μ if and only if it is bounded from \mathfrakh¹m{{\mathfrak{h}}^1}{\mu} to L ¹ μ. Here H ¹ μ denotes the Hardy space introduced in Carbonaro et al. (Ann Sc Norm Super Pisa, 2009), and \mathfrakh¹m{{\mathfrak{h}}^1}{\mu} is defined in Section 4, and is equivalent to a space recently introduced by M. Taylor (J Geom Anal 19(1):137–190, 2009). The case of translation invariant operators on homogeneous trees is also considered.     

Endpoint Results for the Riesz Transform of the Ornstein–Uhlenbeck Operator

Journal of Fourier Analysis and Applications - In this paper we introduce a new atomic Hardy space $$X^1(gamma )$$ adapted to the Gauss measure $$gamma $$ , and prove the boundedness of the first...     

Sharp frequency bounds for eigenfunctions of the Ornstein–Uhlenbeck operator

We prove sharp bounds for the growth rate of eigenfunctions of the Ornstein–Uhlenbeck operator and its natural generalizations. The bounds are sharp even up to lower order terms and have important applications to geometric flows.     

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.     

Higher-order Riesz operators for the Ornstein–Uhlenbeck Semigroup

We prove that the second-order Riesz transforms associated to the Ornstein–Uhlenbeck semigroup are weak type (1,1) with respect to the Gaussian measure in finite dimension. We also show that they are given by a principal value integral plus a constant multiple of the identity. For the Riesz transforms of order three or higher, we present a counterexample showing that the weak type (1,1) estimate fails.     

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 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.     

Principles of Large Deviations for the Empirical Processes of the Ornstein–Uhlenbeck Process

The present paper deals with principles of large deviations for the empirical processes of the Ornstein–Uhlenbeck process. One such principle due to Donsker and Varadhan is well known. It uses as underlying space C(, ^d ) endowed with the topology of uniform convergence on compact sets. The principles of large deviations proved in the present paper use as underlying spaces appropriate subspaces of C(, ^d ) endowed with weighted supremum norms. These principles are natural generalizations of the principle of Donsker and Varadhan.     

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.     

Parameter estimation for the discretely observed fractional Ornstein–Uhlenbeck process and the Yuima R package

This paper proposes consistent and asymptotically Gaussian estimators for the parameters $\lambda , \sigma $ and $H$ of the discretely observed fractional Ornstein–Uhlenbeck process solution of the stochastic differential equation $d Y_t = -\lambda Y_t dt + \sigma d W_t^H$ , where $(W_t^H, t\ge 0)$ is the fractional Brownian motion. For the estimation of the drift $\lambda $ , the results are obtained only in the case when $\frac{1}{2} < H < \frac{3}{4}$ . This paper also provides ready-to-use software for the R statistical environment based on the YUIMA package.