non-Gaussian Ornstein Uhlenbeck Processes Research Articles

Generalized Wiener–Hermite integrals and rough non-Gaussian Ornstein–Uhlenbeck process

We discuss several properties of the generalized Hermite process, which is a non-Gaussian self-similar processes with self-similarity index belonging to the whole interval . We also define Wiener integral with respect to this process and as an application, we define and study its associated Ornstein–Uhlenbeck process.

Application of iterated filtering to stochastic volatility models based on non-Gaussian Ornstein-Uhlenbeck process

Abstract Barndorff-Nielsen and Shephard (2001) proposed a class of stochastic volatility models in which the volatility follows the Ornstein–Uhlenbeck process driven by a positive Levy process without the Gaussian component. The parameter estimation of these models is challenging because the likelihood function is not available in a closed-form expression. A large number of estimation techniques have been proposed, mainly based on Bayesian inference. The main aim of the paper is to present an application of iterated filtering for parameter estimation of such models. Iterated filtering is a method for maximum likelihood inference based on a series of filtering operations, which provide a sequence of parameter estimates that converges to the maximum likelihood estimate. An application to S&P500 index data shows the model perform well and diagnostic plots for iterated filtering ensure convergence iterated filtering to maximum likelihood estimates. Empirical application is accompanied by a simulation study that confirms the validity of the approach in the case of Barndorff-Nielsen and Shephard’s stochastic volatility models.

Adaptive Robust Efficient Methods for Periodic Signal Processing Observed with Colours Noises

In this paper, we consider the problem of robust adaptive efficient estimating a periodic signal observed in the transmission channel with the dependent noise defined by non-Gaussian Ornstein-Uhlenbeck processes with unknown correlation properties. Adaptive model selection procedures, based on the shrinkage weighted least squares estimates, are proposed. The comparison between shrinkage and least squares methods is studied and the advantages of the shrinkage methods are analyzed. Estimation properties for proposed statistical algorithms are studied on the basis of the robust mean square accuracy defined as the maximum mean square estimation error over all possible values of unknown noise parameters. Sharp oracle inequalities for the robust risks have been obtained. The robust efficiency of the model selection procedure has been established.

Lévy-driven non-Gaussian Ornstein–Uhlenbeck processes for degradation-based reliability analysis

ABSTRACTWe use Lévy subordinators and non-Gaussian Ornstein–Uhlenbeck processes to model the evolution of degradation with random jumps. The superiority of our models stems from the flexibility of such processes in the modeling of stylized features of degradation data series such as jumps, linearity/nonlinearity, symmetry/asymmetry, and light/heavy tails. Based on corresponding Fokker–Planck equations, we derive explicit results for the reliability function and lifetime moments in terms of Laplace transforms, represented by Lévy measures. Numerical experiments are used to demonstrate that our general models perform well and are applicable for analyzing a large number of degradation phenomena. More important, they provide us with a new methodology to deal with multi-degradation processes under dynamicenvironments.

GENERALIZED BARNDORFF-NIELSEN AND SHEPHARD MODEL AND DISCRETELY MONITORED OPTION PRICING

This paper proposes a generalization of the Barndorff-Nielsen and Shephard model, in which the log return on an asset is governed by a Lévy process with stochastic volatility modeled by a non-Gaussian Ornstein–Uhlenbeck process. Under the generalized model, we derive a closed-form expression of the multivariate characteristic function of the intertemporal joint distribution of the underlying log return. Then, we also investigate asymptotic behavior of the log return and its variance. Moreover, we evaluate discretely monitored path-dependent derivatives such as geometric Asian, forward start, barrier, fade-in, and lookback options as well as European options.

Robust model selection for a semimartingale continuous time regression from discrete data

The paper considers the problem of estimating a periodic function in a continuous time regression model observed under a general semimartingale noise with an unknown distribution in the case when continuous observation cannot be provided and only discrete time measurements are available. Two specific types of noises are studied in detail: a non-Gaussian Ornstein–Uhlenbeck process and a time-varying linear combination of a Brownian motion and compound Poisson process. We develop new analytical tools to treat the adaptive estimation problems from discrete data. A lower bound for the frequency sampling, needed for the efficiency of the procedure constructed by discrete observations, has been found. Sharp non-asymptotic oracle inequalities for the robust quadratic risk have been derived. New convergence rates for the efficient procedures have been obtained. An example of the regression with a martingale noise exhibits that the minimax robust convergence rate may be both higher or lower as compared with the minimax rate for the “white noise” model. The results of Monte-Carlo simulations are given.

Generalized Barndorff-Nielsen and Shephard Model and Discretely Monitored Option Pricing

This paper proposes a generalization of the Generalized Barndorff-Nielsen and Shephard model, in which the log return on an asset price is governed by a Levy process with stochastic volatility modeled by a non-Gaussian Ornstein-Uhlenbeck process. Under the generalized model, we derive a closed-form expression for the multivariate characteristic function of the intertemporal joint distribution of the underlying log return. Then, we also investigate asymptotic behavior of the log return and its variance. Moreover, we evaluate discretely monitored path-dependent derivatives such as geometric Asian, forward start, barrier, fade-in, and lookback options as well as European options.

Ornstein–Uhlenbeck processes for geophysical data analysis

In this work a three parameter stochastic process, termed the Gamma-Ornstein–Uhlenbeck process, has been implemented to analyze geophysical data. Such non-Gaussian Ornstein–Uhlenbeck processes offer the possibility of capturing important distributional deviations from Gaussianity and make the model flexible of dependence structures. It is shown that the Gamma-Ornstein–Uhlenbeck process is a possible candidate for earthquake data modeling. Rigorous regression analysis is provided and based on that the first-passage times are computed for different sets of data. It is shown that this model may be used to estimate parameters related to some major events—namely major earthquakes.

Modelling the Variance Risk Premium of Equity Indices: The Role of Dependence and Contagion

Understanding variance risk is of key importance in mathematical finance since it affects risk management, asset allocation and derivative pricing. Variance risk is priced in financial markets by the so-called variance risk premium (VRP), which refers to the premium demanded for holding assets whose variance is exposed to stochastic shocks.This paper identifies a new modelling framework for equity indices and presents for the first time explicit analytical formulas for their VRP in a multivariate stochastic volatility setting, which includes multivariate non-Gaussian Ornstein-Uhlenbeck processes and Wishart processes. Moreover, we propose to incorporate contagion within the equity index via a multivariate Hawkes process and find that the resulting dynamics of the VRP represent a convincing alternative to the models studied in the literature up to date.We show that our new model can explain the key stylised facts of both equity indices and individual assets and their corresponding VRP, while popular (multivariate) stochastic volatility models fail.We finally prove the existence of a structure-preserving risk neutral measure for our model. In particular, we establish the class of equivalent probabilities that preserve the self-affecting structure of the Hawkes process.

A truncated estimation method with guaranteed accuracy

This paper presents a truncated estimation method of ratio type functionals by dependent sample of finite size. This method makes it possible to obtain estimators with guaranteed accuracy in the sense of the \(L_m\)-norm, \(m\ge 2\). As an illustration, the parametric and non-parametric estimation problems on a time interval of a fixed length are considered. In particular, parameters of linear (autoregressive) and non-linear discrete-time processes are estimated. Moreover, the parameter estimation problem of non-Gaussian Ornstein-Uhlenbeck process by discrete-time observations and the estimation problem of a multivariate logarithmic derivative of a noise density of an autoregressive process with guaranteed accuracy are solved. In addition to non-asymptotic properties, the limit behavior of presented estimators is investigated. It is shown that all the truncated estimators have asymptotic properties of basic estimators. In particular, the asymptotic efficiency in the mean square sense of the truncated estimator of the dynamic parameter of a stable autoregressive process is established.

Efficient robust nonparametric estimation in a semimartingale regression model

Dans cette article nous considérons le problème d’estimation robuste d’une fonction périodique dans un modèle de régression en temps continu avec un bruit dépendant décrit par une semi martingale carrée intégrable de distribution inconnue. Un exemple de ce bruit est un processus d’Ornstein–Uhlenbeck non gaussien avec sauts (voir (J. R. Stat. Soc. Ser. B Stat. Methodol. 63 (2001) 167–241), (Ann. Appl. Probab. 18 (2008) 879–908)). Nous proposons une procédure adaptative de sélection de modèle basée sur les estimateurs des moindres carrés pondérés. Sous des conditions générales sur les deux premiers moments de la distribution du bruit, des inégalités d’Oracle non asymptotiques pointues pour des risques quadratiques robustes sont obtenues et l’efficacité robuste est établie. Nous avons établi aussi que dans le cas du processus d’Ornstein–Uhlenbeck non Gaussian, la borne inférieure pour le risque quadratique robuste est donnée par la limite de l’intensité du bruit quand la fréquence tend vers l’infini. Nous donnons un exemple d’un modèle de régression avec un bruit martingale où la vitesse de convergence du risque quadratique devient plus lente si l’intensité du bruit tend vers l’infini.

Mean–variance portfolio selection based on a generalized BNS stochastic volatility model

We study a mean–variance portfolio selection problem via optimal feedback control based on a generalized Barndorff-Nielsen and Shephard stochastic volatility model, where an investor trades in a generalized Black–Scholes market. The random coefficients of the market are driven by non-Gaussian Ornstein–Uhlenbeck processes that are independent of the underlying multi-dimensional Brownian motion. Our contribution is to explicitly compute and justify optimal portfolios over an admissible set that is large enough to cover some important classes of strategies such as the class of feedback controls of Markov type. Concretely, the mean–variance efficient portfolios and efficient frontiers are explicitly calculated through the method of generalized linear-quadratic control and explicitly constructed solutions to three integro-partial differential equations under a quite mild condition that only requires one stock whose appreciation-rate process is different from the interest-rate process. Related minimum variance issue is also addressed via our main results.

The Ornstein–Uhlenbeck Dirichlet process and other time-varying processes for Bayesian nonparametric inference

This paper introduces a new class of time-varying, measure-valued stochastic processes for Bayesian nonparametric inference. The class of priors is constructed by normalising a stochastic process derived from non-Gaussian Ornstein–Uhlenbeck processes and generalises the class of normalised random measures with independent increments from static problems. Some properties of the normalised measure are investigated. A particle filter and MCMC schemes are described for inference. The methods are applied to an example in the modelling of financial data.

Joint analysis and estimation of stock prices and trading volume in Barndorff-Nielsen and Shephard stochastic volatility models

We introduce a variant of the Barndorff-Nielsen and Shephard stochastic volatility model where the non-Gaussian Ornstein–Uhlenbeck process describes some measure of trading intensity like trading volume or number of trades instead of unobservable instantaneous variance. We develop an explicit estimator based on martingale estimating functions in a bivariate model that is not a diffusion, but admits jumps. It is assumed that both the quantities are observed on a discrete grid of fixed width, and the observation horizon tends to infinity. We show that the estimator is consistent and asymptotically normal and give explicit expressions of the asymptotic covariance matrix. Our method is illustrated by a finite sample experiment and a statistical analysis of IBM™ stock from the New York Stock Exchange and Microsoft Corporation™ stock from Nasdaq during a history of five years.

Indirect inference methods for stochastic volatility models based on non-Gaussian Ornstein–Uhlenbeck processes

An indirect inference method is implemented for a class of stochastic volatility models for financial data based on non-Gaussian Ornstein–Uhlenbeck (OU) processes. First, a quasi-likelihood estimator is derived from an approximative Gaussian state space representation of the OU model. Next, data are simulated from the OU model for given parameter values. The indirect inference estimator is then obtained by minimizing, in a weighted mean squared error sense, the score vector of the quasi-likelihood function for the simulated data, when this score vector is evaluated at the quasi-likelihood estimator obtained from the real data. The method is applied to Euro/Norwegian krone (NOK) and US Dollar/NOK daily exchange rate data. A simulation study reveals that the quasi-likelihood estimator may have a large bias even in large samples, but that the indirect inference estimator substantially reduces this bias. The accompanying R-package, which interfaces C++ code, is documented and can be downloaded.

Bayesian inference with stochastic volatility models using continuous superpositions of non-Gaussian Ornstein–Uhlenbeck processes

Continuous superpositions of Ornstein–Uhlenbeck processes are proposed as a model for asset return volatility. An interesting class of continuous superpositions is defined by a Gamma mixing distribution which can define long memory processes. In contrast, previously studied discrete superpositions cannot generate this behaviour. Efficient Markov chain Monte Carlo methods for Bayesian inference are developed which allow the estimation of such models with leverage effects. The continuous superposition model is applied to both stock index and exchange rate data. The continuous superposition model is compared with a two-component superposition on the daily Standard and Poor’s 500 index from 1980 to 2000.

Optimal investment and consumption in a Black–Scholes market with Lévy-driven stochastic coefficients

In this paper, we investigate an optimal investment and consumption problem for an investor who trades in a Black–Scholes financial market with stochastic coefficients driven by a non-Gaussian Ornstein–Uhlenbeck process. We assume that an agent makes investment and consumption decisions based on a power utility function. By applying the usual separation method in the variables, we are faced with the problem of solving a nonlinear (semilinear) first-order partial integro-differential equation. A candidate solution is derived via the Feynman–Kac representation. By using the properties of an operator defined in a suitable function space, we prove uniqueness and smoothness of the solution. Optimality is verified by applying a classical verification theorem.

Inference with non-Gaussian Ornstein–Uhlenbeck processes for stochastic volatility

Continuous-time stochastic volatility models are becoming an increasingly popular way to describe moderate and high-frequency financial data. Barndorff-Nielsen and Shephard (2001a) proposed a class of models where the volatility behaves according to an Ornstein–Uhlenbeck (OU) process, driven by a positive Levy process without Gaussian component. These models introduce discontinuities, or jumps, into the volatility process. They also consider superpositions of such processes and we extend that to the inclusion of a jump component in the returns. In addition, we allow for leverage effects and we introduce separate risk pricing for the volatility components. We design and implement practically relevant inference methods for such models, within the Bayesian paradigm. The algorithm is based on Markov chain Monte Carlo (MCMC) methods and we use a series representation of Levy processes. MCMC methods for such models are complicated by the fact that parameter changes will often induce a change in the distribution of the representation of the process and the associated problem of overconditioning. We avoid this problem by dependent thinning methods. An application to stock price data shows the models perform very well, even in the face of data with rapid changes, especially if a superposition of processes with different risk premiums and a leverage effect is used.

Asymptotic expansion for Barndorff-Nielsen and Shephard's stochastic volatility model

With the help of a general methodology of asymptotic expansions for mixing processes, we obtain the Edgeworth expansion for log-returns of a stock price process in Barndorff-Nielsen and Shephard's stochastic volatility model, in which the latent volatility process is described by a stationary non-Gaussian Ornstein–Uhlenbeck process (OU process) with invariant selfdecomposable distribution on R + . The present result enables us to simultaneously explain non-Gaussianity for short time-lags as well as approximate Gaussianity for long time-lags. The Malliavin calculus formulated by Bichteler, Gravereaux and Jacod for processes with jumps and the exponential mixing property of the OU process play substantial roles in order to ensure a conditional type Cramér condition under a certain truncation. Owing to several inherent properties of OU processes, the regularity conditions for the expansions can be verified without any difficulty, and the coefficients of the expansions up to any order can be explicitly computed.

THE NORMAL INVERSE GAUSSIAN DISTRIBUTION AND SPOT PRICE MODELLING IN ENERGY MARKETS

We model spot prices in energy markets with exponential non-Gaussian Ornstein–Uhlenbeck processes. We generalize the classical geometric Brownian motion and Schwartz' mean-reversion model by introducing Lévy processes as the driving noise rather than Brownian motion. Instead of modelling the spot price dynamics as the solution of a stochastic differential equation with jumps, it is advantageous from a statistical point of view to model the price process directly. Imposing the normal inverse Gaussian distribution as the statistical model for the Lévy increments, we obtain a superior fit compared to the Gaussian model when applied to spot price data from the oil and gas markets. We also discuss the problem of pricing forwards and options and outline how to find the market price of risk in an incomplete market.