Uhlenbeck Type Processes Research Articles

Tail Behavior and Almost Sure Growth Rate of Superpositions of Ornstein–Uhlenbeck-type Processes

Tail Behavior and Almost Sure Growth Rate of Superpositions of Ornstein–Uhlenbeck-type Processes

Scaling limits for a class of regular [formula omitted]-coalescents

Let Nt(n) denote the number of blocks in a Ξ-coalescent restricted to a sample of size n∈N after time t≥0. Under the assumption of a certain curvature condition on a function well-known from the literature, we prove the existence of sequences (v(n,t))n∈N for which (logNt(n)−logv(n,t))t≥0 converges to an Ornstein–Uhlenbeck type process as n→∞. The curvature condition is intrinsically related to the behavior of Ξ near the origin. The method of proof is to show the uniform convergence of the associated generators. Via Siegmund duality an analogous result for the fixation line is proven. Several examples are studied.

Pricing European Options under Stochastic Volatility Models: Case of Five-Parameter Variance-Gamma Process

The paper builds a Variance-Gamma (VG) model with five parameters: location (μ), symmetry (δ), volatility (σ), shape (α), and scale (θ); and studies its application to the pricing of European options. The results of our analysis show that the five-parameter VG model is a stochastic volatility model with a Γ(α,θ) Ornstein–Uhlenbeck type process; the associated Lévy density of the VG model is a KoBoL family of order ν=0, intensity α, and steepness parameters δσ2−δ2σ4+2θσ2 and δσ2+δ2σ4+2θσ2; and the VG process converges asymptotically in distribution to a Lévy process driven by a normal distribution with mean (μ+αθδ) and variance α(θ2δ2+σ2θ). The data used for empirical analysis were obtained by fitting the five-parameter Variance-Gamma (VG) model to the underlying distribution of the daily SPY ETF data. Regarding the application of the five-parameter VG model, the twelve-point rule Composite Newton–Cotes Quadrature and Fractional Fast Fourier (FRFT) algorithms were implemented to compute the European option price. Compared to the Black–Scholes (BS) model, empirical evidence shows that the VG option price is underpriced for out-of-the-money (OTM) options and overpriced for in-the-money (ITM) options. Both models produce almost the same option pricing results for deep out-of-the-money (OTM) and deep-in-the-money (ITM) options.

Langevin original approach and Ornstein–Uhlenbeck-type processes

This paper applies Langevin idea to describe the Brownian motion of a particle characterized by an Ornstein–Uhlenbeck-type process. The original and clever method proposed by Langevin is based on Newton’s second law plus a fluctuating force whose solution for the mean square displacement consists in separating the fluctuating force from his equation to obtain a deterministic equation for the relevant physical variable. In this work the Langevin original idea is applied to calculate the mean square velocity for a field free particle case; then it is extended for a charged particle in a constant magnetic field. In a similar way, the strategy is also applied to calculate the mean square displacement for a Brownian harmonic oscillator in the overdamped regime, and also when a magnetic field is present. In particular, it is shown in the field free case that Langevin’s original strategy leads to the same results as those obtained using the statistical properties of a Gaussian white noise. All the theoretical results are compared with both the numerical simulation of the Langevin equation, and numerical solution of the corresponding deterministic differential equations.

Detecting time-changes in $ PM_{10} $ during Covid pandemic by means of an Ornstein Uhlenbeck type process.

Particulate matter with 10 micrometers or less in diameter ($ PM_{10} $) from several italian cities is modeled by means of a non homogeneous Ornstein Uhlenbeck process. Such model includes two deterministic time dependent functions in the infinitesimal moments to describe the presence of exogeneous terms in the typical dynamics of the phenomenon. An iterative estimating procedure combining the maximum likelihood estimation and a generalized method of moments is provided. A Quandt Likelihood Ratio test for detecting structural breaks in $ PM_{10} $ data, in the period from 1st January 2020 to 8th July 2020 which includes the first lockdown due to Covid pandemic, confirms the presence of time-changes. These results show that the lockdown made the air once again cleaner. It is then shown that our model and the associated estimation procedure, while not explicitly contemplating the presence of structural breaks in the time series, implicitly incorporates them in the time dependence of the functions in the infinitesimal moments of the underlying process.

An optimal Gauss–Markov approximation for a process with stochastic drift and applications

We consider a linear stochastic differential equation with stochastic drift. We study the problem of approximating the solution of such equation through an Ornstein–Uhlenbeck type process, by using direct methods of calculus of variations. We show that general power cost functionals satisfy the conditions for existence and uniqueness of the approximation. We provide some examples of general interest and we give bounds on the goodness of the corresponding approximations. Finally, we focus on a model of a neuron embedded in a simple network and we study the approximation of its activity, by exploiting the aforementioned results.

Inferring time non-homogeneous Ornstein Uhlenbeck type stochastic process

A generalization of the classical Ornstein Uhlenbeck diffusion process including some deterministic time dependent functions in the infinitesimal moments is considered. The inference based on discrete sampling in time is provided by means of an iterative procedure that, in each step, combines the classical maximum likelihood estimation and a generalized method of moments. The validity of the suggested procedure is evaluated via a simulation study by considering several infinitesimal moments for the Ornstein Uhlenbeck type process and taking different sample size. Finally, an application to PM10 daily concentration in Turin metropolitan area in Italy is discussed.

A growth-fragmentation model related to Ornstein–Uhlenbeck type processes

Growth-fragmentation processes describe systems of particles in which each particle may grow larger or smaller, and divide into smaller ones as time proceeds. Unlike previous studies, which have focused mainly on the self-similar case, we introduce a new type of growth-fragmentation which is closely related to L\'evy driven Ornstein-Uhlenbeck type processes. Our model can be viewed as a generalization of compensated fragmentation processes introduced by Bertoin, or the stochastic counterpart of a family of growth-fragmentation equations. We establish a convergence criterion for a sequence of such growth-fragmentations. We also prove that, under certain conditions, this system fulfills a law of large numbers.

The Multifaceted Behavior of Integrated supOU Processes: The Infinite Variance Case

The so-called supOU processes, namely the superpositions of Ornstein–Uhlenbeck type processes, are stationary processes for which one can specify separately the marginal distribution and the temporal dependence structure. They can have finite or infinite variance. We study the limit behavior of integrated infinite variance supOU processes adequately normalized. Depending on the specific circumstances, the limit can be fractional Brownian motion but it can also be a process with infinite variance, a Levy stable process with independent increments or a stable process with dependent increments. We show that it is even possible to have infinite variance integrated supOU processes converging to processes whose moments are all finite. A number of examples are provided.

The unusual properties of aggregated superpositions of Ornstein–Uhlenbeck type processes

Superpositions of Ornstein–Uhlenbeck type (supOU) processes form a rich class of stationary processes with a flexible dependence structure. The asymptotic behavior of the integrated and partial sum supOU processes can be, however, unusual. Their cumulants and moments turn out to have an unexpected rate of growth. We identify the property of fast growth of moments or cumulants as intermittency. Many proofs are given in a supplemental article (Grahovac, Leonenko, Sikorskii and Taqqu (2018)).

Limit theorems, scaling of moments and intermittency for integrated finite variance supOU processes

Superpositions of Ornstein–Uhlenbeck type (supOU) processes provide a rich class of stationary stochastic processes for which the marginal distribution and the dependence structure may be modeled independently. We show that they can also display intermittency, a phenomenon affecting the rate of growth of moments. To do so, we investigate the limiting behavior of integrated supOU processes with finite variance. After suitable normalization four different limiting processes may arise depending on the decay of the correlation function and on the characteristic triplet of the marginal distribution. To show that supOU processes may exhibit intermittency, we establish the rate of growth of moments for each of the four limiting scenarios. The rate change indicates that there is intermittency, which is expressed here as a change-point in the asymptotic behavior of the absolute moments.

Statistical Analysis of the Mixed Fractional Ornstein--Uhlenbeck Process

This paper addresses the problem of estimating the drift parameter of the Ornstein--Uhlenbeck-type process driven by the sum of independent standard and fractional Brownian motions. With the help of some recent results on the canonical representation and spectral structure of mixed processes, the maximum likelihood estimator is shown to be consistent and asymptotically normal in the large-sample limit.

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.

A non-Gaussian Ornstein–Uhlenbeck model for pricing wind power futures

ABSTRACTThe recent introduction of wind power futures written on the German wind power production index has brought with it new interesting challenges in terms of modelling and pricing. Some particularities of this product are the strong seasonal component embedded in the underlying, the fact that the wind index is bounded from both above and below and also that the futures are settled against a synthetically generated spot index. Here, we consider the non-Gaussian Ornstein–Uhlenbeck type processes proposed by Barndorff-Nielsen and Shephard in the context of modelling the wind power production index. We discuss the properties of the model and estimation of the model parameters. Further, the model allows for an analytical formula for pricing wind power futures. We provide an empirical study, where the model is calibrated to 37 years of German wind power production index that is synthetically generated assuming a constant level of installed capacity. Also, based on 1 year of observed prices for wind power futures with different delivery periods, we study the market price of risk. Generally, we find a negative risk premium whose magnitude decreases as the length of the delivery period increases. To further demonstrate the benefits of our proposed model, we address the pricing of European options written on wind power futures, which can be achieved through Fourier techniques.

Statistical analysis of the mixed fractional Ornstein-Uhlenbeck process

Рассматривается задача оценивания коэффициента сноса процесса типа Орнштейна-Уленбека, управляемого суммой независимых стандартного и фрактального броуновских движений. С использованием недавних результатов о каноническом представлении и спектральной структуре смешанных процессов показано, что оценка максимального правдоподобия является состоятельной и асимптотически нормальной, когда время наблюдения стремится к бесконечности.

Intermittency of Superpositions of Ornstein–Uhlenbeck Type Processes

The phenomenon of intermittency has been widely discussed in physics literature. This paper provides a model of intermittency based on L\'evy driven Ornstein-Uhlenbeck (OU) type processes. Discrete superpositions of these processes can be constructed to incorporate non-Gaussian marginal distributions and long or short range dependence. While the partial sums of finite superpositions of OU type processes obey the central limit theorem, we show that the partial sums of a large class of infinite long range dependent superpositions are intermittent. We discuss the property of intermittency and behavior of the cumulants for the superpositions of OU type processes.

INVESTOR'S SENTIMENT IN MULTI-AGENT MODEL OF THE CONTINUOUS DOUBLE AUCTION

We introduce and treat rigorously a new multi-agent model of the continuous double auction or in other words the order book (OB). It is designed to explain collective behavior of the market when new information affecting the market arrives. The novel feature of the model is two additional slow changing parameters, the so-called sentiment functions. These sentiment functions measure the conception of the fair price of two groups of investors, namely, bulls and bears. Our model specifies differential equations for the time evolution of sentiment functions and constitutes a nonlinear Markov process which exhibits long-term correlations. We explain the intuition behind equations for sentiment functions and present numerical simulations which show that the behavior of our model is similar to the behavior of the real market. We also obtain a diffusion limit of the model, the Ornstein–Uhlenbeck type process with variable volatility. The volatility is proportional to the difference of opinions of bulls and bears about the fair price of a security.

Efficient maximum likelihood estimation for Lévy-driven Ornstein–Uhlenbeck processes

We consider the problem of efficient estimation of the drift parameter of an Ornstein–Uhlenbeck type process driven by a Lévy process when high-frequency observations are given. The estimator is constructed from the time-continuous likelihood function that leads to an explicit maximum likelihood estimator and requires knowledge of the continuous martingale part. We use a thresholding technique to approximate the continuous part of the process. Under suitable conditions, we prove asymptotic normality and efficiency in the Hájek–Le Cam sense for the resulting drift estimator. Finally, we investigate the finite sample behavior of the method and compare our approach to least squares estimation.

Ultimate efficiency of experimental designs for Ornstein–Uhlenbeck type processes

For processes governed by linear Itō stochastic differential equations of the form dX(t)=[a(t)+b(t)X(t)]dt+σ(t)dW(t), we discuss the existence of optimal sampling designs with strictly increasing sampling times. We derive an asymptotic Fisher information matrix, which we take as a reference in assessing the quality of the finite-point sampling designs. The results are extended to a broader class of Itō stochastic differential equations. We give an example based on the Gompertz tumour growth law.

A Generalized Hyperbolic model for a risky asset with dependence

We present a construction of the Generalized Hyperbolic (GH) subordinator model for a risky asset with dependence. The construction of the subordinator (activity time) process is implemented via superpositions of Ornstein–Uhlenbeck type processes driven by Lévy noise. It unifies, on the basis of self-decomposability of the Generalized Inverse Gaussian (GIG) distribution, the construction of the various special cases of the GH subordinator class, such as the Variance Gamma, normal inverse Gaussian, hyperbolic and, especially, t distributions. An option pricing formula for the proposed model is derived.