Stable Ornstein Uhlenbeck Research Articles

A remark on exact simulation of tempered stable Ornstein–Uhlenbeck processes

Abstract Qu, Dassios, and Zhao (2021) suggested an exact simulation method for tempered stable Ornstein–Uhlenbeck processes, but their algorithms contain some errors. This short note aims to correct their algorithms and conduct some numerical experiments.

Ergodicity bounds for stable Ornstein-Uhlenbeck systems in Wasserstein distance with applications to cutoff stability.

This article establishes cutoff stability also known as abrupt thermalization for generic multidimensional Hurwitz stable Ornstein-Uhlenbeck systems with (possibly degenerate) Lévy noise at fixed noise intensity. The results are based on several ergodicity quantitative lower and upper bounds some of which make use of the recently established shift linearity property of the Wasserstein-Kantorovich-Rubinstein distance by the authors. It covers such irregular systems like Jacobi chains and more general networks of coupled harmonic oscillators with a heat bath (including Lévy excitations) at constant temperature on the outer edges and the so-called Brownian gyrator.

Creeping of Lévy processes through curves

A Lévy process is said to creep through a curve if, at its first passage time across this curve, the process reaches it with positive probability. We first study this property for bivariate subordinators. Given the graph {(t,f(t)):t≥0} of any continuous, non increasing function f such that f(0)>0, we give an expression of the probability that a bivariate subordinator (Y,Z) issued from 0 creeps through this graph in terms of its renewal function and the drifts of the components Y and Z. We apply this result to the creeping probability of any real Lévy process through the graph of any continuous, non increasing function at a time where the process also reaches its past supremum. This probability involves the density of the renewal function of the bivariate upward ladder process as well as its drift coefficients. We also investigate the case of Lévy processes conditioned to stay positive creeping at their last passage time below the graph of a function. Then we provide some examples and we give an application to the probability of creeping through fixed levels by stable Ornstein-Uhlenbeck processes. We also raise a couple of open questions along the text.

Fast simulation of tempered stable Ornstein–Uhlenbeck processes

Constructing Lévy-driven Ornstein–Uhlenbeck processes is a task closely related to the notion of self-decomposability. In particular, their transition laws are linked to the properties of what will be hereafter called the a-remainder of their self-decomposable stationary laws. In the present study we fully characterize the Lévy triplet of these a-remainders and we provide a general framework to deduce the transition laws of the finite variation Ornstein–Uhlenbeck processes associated with tempered stable distributions. We focus finally on the subclass of the exponentially-modulated tempered stable laws and we derive the algorithms for an exact generation of the skeleton of Ornstein–Uhlenbeck processes related to such distributions, with the further advantage of adopting procedures which are tens of times faster than those already available in the existing literature.

Explicit representation of characteristic function of tempered α‐stable Ornstein–Uhlenbeck process

The Ornstein–Uhlenbeck process is a fundamental tool in various applications ranging from finance where it is known under Vasiček process name to fractional diffusion in the context, for instance, Klein–Kramers dynamics. In this paper, we define the Ornstein–Uhlenbeck process driven by a symmetric tempered α‐stable process. We investigate its properties in the non‐stationary case and its long time limit. Moreover, we provide an explicit representation of its characteristic function in terms of special functions, that is, generalized hypergeometric function and its special case hypergeometric Gauss function. Thus, using probabilistic arguments, we provide a new identity between those two functions. Our calculations are also confirmed by numerical simulations of this new process. We discuss the Fokker–Planck type equation describing the probability density function of our model as a solution of the fractional differential equation with the so‐called tempered fractional derivative.

Multidimensional parameter estimation of heavy‐tailed moving averages

AbstractIn this article we present a parametric estimation method for certain multiparameter heavy‐tailed Lévy‐driven moving averages. The theory relies on recent multivariate central limit theorems obtained via Malliavin calculus on Poisson spaces. Our minimal contrast approach is related to previous papers, which propose to use the marginal empirical characteristic function to estimate the one‐dimensional parameter of the kernel function and the stability index of the driving Lévy motion. We extend their work to allow for a multiparametric framework that in particular includes the important examples of the linear fractional stable motion, the stable Ornstein–Uhlenbeck process, certain CARMA(2, 1) models, and Ornstein–Uhlenbeck processes with a periodic component among other models. We present both the consistency and the associated central limit theorem of the minimal contrast estimator. Furthermore, we demonstrate numerical analysis to uncover the finite sample performance of our method.

Tempered Stable Ornstein–Uhlenbeck Model for River Discharge Time Series with Its Application to Dissolved Silicon Load Analysis

We identify stochastic process models describing the time series of inflow and outflow discharges of Obara Dam in Hii River, Japan. These models are based on tempered stable Ornstein–Uhlenbeck (TSOU) processes that have not been utilized in hydrological analysis but can capture both large and small fluctuation of the time series data. In addition, the models can be exactly simulated in a statistical sense by utilizing a recent tailored discretization algorithm, serving as efficient stochastic tools. We show that the identified models accurately reproduce statistical moments of the time series data and probability density functions. Based on the mathematical framework of backward stochastic differential equation (BSDE), the identified model is applied to a unique dynamic stochastic analysis of dissolved silicon (DSi) load flowing into the reservoir associated with Obara Dam. We thus contribute to the first application of TSOU processes and a BSDE to hydrological analysis.

Maximum likelihood estimation for symmetric α-stable Ornstein–Uhlenbeck processes

In this paper, we consider the maximum likelihood estimation for the symmetric [Formula: see text]-stable Ornstein–Uhlenbeck (S[Formula: see text]S-OU) processes based on discrete observations. Since the closed-form expression of maximum likelihood function is hard to obtain in the Lévy case, we choose a mixture of Cauchy and Gaussian distribution to approximate the probability density function (PDF) of the S[Formula: see text]S distribution. By means of transition function and Laplace transform, we construct an explicit approximate sequence of likelihood function, which converges to the likelihood function of S[Formula: see text]S distribution. Based on the approximation of likelihood function we give an algorithm for computing maximum likelihood estimation. We also numerically simulate some experiments which demonstrate the accuracy and stability of the proposed estimator.

Schauder estimates for degenerate stable Kolmogorov equations

We provide here global Schauder-type estimates for a chain of integro-partial differential equations (IPDE) driven by a degenerate stable Ornstein-Uhlenbeck operator possibly perturbed by a deterministic drift, when the coefficients lie in some suitable anisotropic Hölder spaces. Our approach mainly relies on a perturbative method based on forward parametrix expansions and, due to the low regularizing properties on the degenerate variables and to some integrability constraints linked to the stability index, it also exploits duality results between appropriate Besov Spaces. In particular, our method also applies in some super-critical cases. Thanks to these estimates, we show in addition the well-posedness of the considered IPDE in a suitable functional space.

On the simulation of general tempered stable Ornstein–Uhlenbeck processes

ABSTRACTWe give an explicit representation for the transition law of a tempered stable Ornstein–Uhlenbeck process and use it to develop a rejection sampling algorithm for exact simulation of increments from this process. Our results apply to general classes of both univariate and multivariate tempered stable distributions and contain a number of previously studied results as special cases.

A Berry–Esseén theorem for partial sums of functionals of heavy-tailed moving averages

In this paper we obtain Berry–Esseén bounds on partial sums of functionals of heavy-tailed moving averages, including the linear fractional stable noise, stable fractional ARIMA processes and stable Ornstein–Uhlenbeck processes. Our rates are obtained for the Wasserstein and Kolmogorov distances, and depend strongly on the interplay between the memory of the process, which is controlled by a parameter $\alpha $, and its tail-index, which is controlled by a parameter $\beta $. In fact, we obtain the classical $1/\sqrt {n}$ rate of convergence when the tails are not too heavy and the memory is not too strong, more precisely, when $\alpha \beta >3$ or $\alpha \beta >4$ in the case of Wasserstein and Kolmogorov distance, respectively. Our quantitative bounds rely on a new second-order Poincaré inequality on the Poisson space, which we derive through a combination of Stein’s method and Malliavin calculus. This inequality improves and generalizes a result by Last, Peccati, Schulte [Probab. Theory Relat. Fields 165 (2016)].

Generalized moment estimators for $$\alpha $$-stable Ornstein–Uhlenbeck motions from discrete observations

We study the parameter estimation problem for discretely observed Ornstein–Uhlenbeck processes driven by $$\alpha $$-stable Levy motions. A method of moments via ergodic theory and via sample characteristic functions is proposed to estimate all the parameters involved in the Ornstein–Uhlenbeck processes. We obtain the strong consistency and asymptotic normality of the proposed joint estimators when the sample size $$n \rightarrow \infty $$ while the sampling time step h remains arbitrarily fixed. We also design a procedure to select the grid points in the characteristic functions in certain optimal way for the proposed estimators.

Approximation to Default Swaptions in Reduced-Form Framework: Calibration and Comparison of Intensity Dynamics

We develop an approximation to pricing default swaptions in general intensity models which yields a fast and accurate analytic/semi-analytic solution to the valuation problem. We consider several dynamics for intensity and derive 'extended transforms' for inverse Gaussian- and tempered stable Ornstein-Uhlenbeck processes. Having convenient pricing formulae, we are able to provide the first account of calibration of intensity models to credit options. Further, we extend the model by adding a shift, thus resulting in a tractable model that keeps the intensity positive, matches closely implied volatilities of default swaptions, and fits exactly the CDS spreads.

Infinite Variation Tempered Stable Ornstein–Uhlenbeck Processes with Discrete Observations

We investigate transition law between consecutive observations of Ornstein–Uhlenbeck processes of infinite variation with tempered stable stationary distribution. Thanks to the Markov autoregressive structure, the transition law can be written in the exact sense as a convolution of three random components; a compound Poisson distribution and two independent tempered stable distributions, one with stability index in (0, 1) and the other with index in (1, 2). We discuss simulation techniques for those three random elements. With the exact transition law and proposed simulation techniques, sample paths simulation proves significantly more efficient, relative to the known approximative technique based on infinite shot noise series representation of tempered stable Lévy processes.

Exact simulation of tempered stable Ornstein–Uhlenbeck processes

Based on a representation of a stochastic integral of Ornstein–Uhlenbeck (O–U) type, the exact simulation algorithm of the tempered stable O–U process is given in this paper. The algorithm employs the double rejection method and the general acceptance–rejection technique. The time complexity of the double rejection method is uniformly bounded over all values of the parameter. And the acceptance probability of the acceptance–rejection technique can be improved to as close to 1 as possible. Thus, the implementation of the algorithm is efficient. The performance of the simulation method is evidenced by some empirical results.

Subordinated [formula omitted]-stable Ornstein–Uhlenbeck process as a tool for financial data description

The classical financial models are based on the standard Brownian diffusion-type processes. However, in the exhibition of some real market data (like interest or exchange rates) we observe characteristic periods of constant values. Moreover, in the case of financial data, the assumption of normality is often unsatisfied. In such cases the popular Vasiček model, that is a mathematical system describing the evolution of interest rates based on the Ornstein–Uhlenbeck process, seems not to be applicable. Therefore, we propose an alternative approach based on a combination of the popular Ornstein–Uhlenbeck process with a stable distribution and subdiffusion systems that demonstrate such characteristic behavior. The probability density function of the proposed process can be described by a Fokker–Planck type equation and therefore it can be examined as an extension of the basic Ornstein–Uhlenbeck model. In this paper, we propose the parameters’ estimation method and calibrate the subordinated Vasiček model to the interest rate data.

Exact discrete sampling of finite variation tempered stable Ornstein–Uhlenbeck processes

Exact yet simple simulation algorithms are developed for a wide class of Orn- stein-Uhlenbeck processes with tempered stable stationary distribution of finite varia- tion with the help of their exact transition probability between consecutive time points. Random elements involved can be divided into independent tempered stable and com- pound Poisson distributions, each of which can be simulated in the exact sense through acceptance-rejection sampling, respectively, with stable and gamma proposal distribu- tions. We discuss various alternative simulation methods within our algorithms on the ba- sis of acceptance rate in acceptance-rejection sampling for both high- and low-frequency sampling. Numerical results illustrate their advantage relative to the existing approxima- tive simulation method based on infinite shot noise series representation.

Quantile inference for near-integrated autoregressive time series under infinite variance and strong dependence

Consider a near-integrated time series driven by a heavy-tailed and long-memory noise ε t = ∑ j = 0 ∞ c j η t − j , where { η j } is a sequence of i . i . d random variables belonging to the domain of attraction of a stable law with index α . The limit distribution of the quantile estimate and the semi-parametric estimate of the autoregressive parameters with long- and short-range dependent innovations are established in this paper. Under certain regularity conditions, it is shown that when the noise is short-memory, the quantile estimate converges weakly to a mixture of a Gaussian process and a stable Ornstein–Uhlenbeck (O–U) process while the semi-parametric estimate converges weakly to a normal distribution. But when the noise is long-memory, the limit distribution of the quantile estimate becomes substantially different. Depending on the range of the stable index α , the limit distribution is shown to be either a functional of a fractional stable O–U process or a mixture of a stable process and a stable O–U process. These results indicate that although the quantile estimate tends to be more efficient for infinite variance time series, extreme caution should be exercised in the long-memory situation.

On the Transition Law of Tempered Stable Ornstein–Uhlenbeck Processes

In this paper, a stochastic integral of Ornstein–Uhlenbeck type is represented to be the sum of two independent random variables: one has a tempered stable distribution and the other has a compound Poisson distribution. In distribution, the compound Poisson random variable is equal to the sum of a Poisson-distributed number of positive random variables, which are independent and identically distributed and have a common specified density function. Based on the representation of the stochastic integral, we prove that the transition distribution of the tempered stable Ornstein–Uhlenbeck process is self-decomposable and that the transition density is a C∞-function.

On the Transition Law of Tempered Stable Ornstein–Uhlenbeck Processes

In this paper, a stochastic integral of Ornstein–Uhlenbeck type is represented to be the sum of two independent random variables: one has a tempered stable distribution and the other has a compound Poisson distribution. In distribution, the compound Poisson random variable is equal to the sum of a Poisson-distributed number of positive random variables, which are independent and identically distributed and have a common specified density function. Based on the representation of the stochastic integral, we prove that the transition distribution of the tempered stable Ornstein–Uhlenbeck process is self-decomposable and that the transition density is aC∞-function.