Ornstein-Uhlenbeck Type Research Articles

Fractal Activity Time Models for Risky Asset with Dependence and Generalized Hyperbolic Distributions

Risky asset models with the dependence through fractal activity time are described. The construction of the fractal activity time is implemented via superpositions of Ornstein-Uhlenbeck type processes driven by Lévy noise. The model features both tractable dependence structure and desired marginal distributions of the returns from the generalized hyperbolic class: the Variance Gamma and normal inverse Gaussian. These distributions provide good fit to real financial data. Pricing formulae for the proposed models are derived.

The local time of the Markov processes of Ornstein–Uhlenbeck type

We prove the existence of the local time of the Ornstein–Uhlenbeck type process X={Xt,t∈R+} driven by a general Lévy process. The conditions of the continuity in time variable t and the regularity property of the local time are given, under mild regularity conditions on the driving Lévy process. These results give information about the local properties of sample functions of the process.

Identities for vector fields in the infinitesimal representation of the symplectic group into the Siegel disk of complex symmetric matrices

We discuss the notion of Ornstein–Uhlenbeck operator on a complex manifold endowed with a Kählerian metric. We give the example of the Siegel disk. We consider the infinitesimal holomorphic representation of Sp ( 2 n ) , the symplectic group of order n , into the Siegel disk D n of symmetric complex n × n matrices. Let ρ ( v ) = L ( v ) + β ( v ) I , the first order differential operator on D n associated to the element v in the Lie algebra G of Sp ( 2 n ) . We denote L ( v ) a vector field, β ( v ) a function on D n and β ( v ) I is the operator of multiplication by β ( v ) . We show the existence of a basis ( e k ) in the Lie algebra G and of constants ( a k ) such that the operator ∑ k a k ρ ( e k ) 2 is equal to the multiplication by a constant. The constants ( a k ) can be taken equal to 1 for n 2 + n of them and to −1 for the others. Varying the coefficients in the modular factor of the representation, we obtain Ornstein–Uhlenbeck type operators on D n of the form ∑ k a k ρ ( e k ) L ( e k ) ¯ where L ( e k ) ¯ is the complex conjugate of L ( e k ) . In particular the Kählerian Laplacian on D n is expressed as ∑ k a k L ( e k ) L ( e k ) ¯ . The imaginary part of the vector field ∑ k a k β ( e k ) L ( e k ) ¯ is divergence free for the measure of the holomorphic representation. This extends some of the identities obtained for the Poincaré disk in H. Airault and H. Ouerdiane (2011, 2009) [4] , [3] .

Random matrix models of stochastic integral type for free infinitely divisible distributions

The Bercovici-Pata bijection maps the set of classical infinitely divisible distributions to the set of free infinitely divisible distributions. The purpose of this work is to study random matrix models for free infinitely divisible distributions under this bijection. First, we find a specific form of the polar decomposition for the Levy measures of the random matrix models considered in Benaych-Georges [6] who introduced the models through their laws. Second, random matrix models for free infinitely divisible distributions are built consisting of infinitely divisible matrix stochastic integrals whenever their corresponding classical infinitely divisible distributions admit stochastic integral representations. These random matrix models are realizations of random matrices given by stochastic integrals with respect to matrix-valued Levy processes. Examples of these random matrix models for several classes of free infinitely divisible distributions are given. In particular, it is shown that any free selfdecomposable infinitely divisible distribution has a random matrix model of Ornstein-Uhlenbeck type ∫0∞e−1dΨtd, d ≥ 1, where Ψtd is a d × d matrix-valued Levy process satisfying an Ilog condition.

Hitting half-spaces or spheres by Ornstein–Uhlenbeck type diffusions

The purpose of the paper is to provide a general method for computing the hitting distributions of some regular subsets $D$ for Ornstein–Uhlenbeck type operators of the form ${1\over 2}\varDelta + F\cdot \nabla $, with $F$ bounded and orthogonal to

Option Pricing in Multivariate Stochastic Volatility Models of OU Type

We present a multivariate stochastic volatility model with leverage, which is flexible enough to recapture the individual dynamics as well as the interdependencies between several assets, while still being highly analytically tractable. First, we derive the characteristic function and give conditions that ensure its analyticity and absolute integrability in some open complex strip around zero. Therefore we can use Fourier methods to compute the prices of multiasset options efficiently. To show the applicability of our results, we propose a concrete specification, the Ornstein–Uhlenbeck (OU)–Wishart model, where the dynamics of each individual asset coincide with the popular $\Gamma$-OU Barndorff-Nielsen–Shepard model. This model can be well calibrated to market prices, which we illustrate with an example using options on the exchange rates of some major currencies. Finally, we show that covariance swaps can also be priced in closed form.

General alpha-Wiener bridges

An α-Wiener bridge is a one-parameter generalization of the usual Wiener bridge, where the parameter α > 0 represents a mean reversion force to zero. We generalize the notion of α-Wiener bridges to continuous functions α : [0, T ) → R. We show that if the limit limt↑T α(t) exists and is positive, then a general α-Wiener bridge is in fact a bridge in the sense that it converges to 0 at time T with probability one. Further, under the condition limt↑T α(t) 6= 1 we show that the law of the general α-Wiener bridge cannot coincide with the law of any non time-homogeneous Ornstein-Uhlenbeck type bridge. In case limt↑T α(t) = 1 we determine all the Ornstein-Uhlenbeck type processes from which one can derive the general α-Wiener bridge by conditioning the original Ornstein-Uhlenbeck type process to be in 0 at time T .

THE MULTIVARIATE supOU STOCHASTIC VOLATILITY MODEL

Using positive semidefinite supOU (superposition of Ornstein–Uhlenbeck type) processes to describe the volatility, we introduce a multivariate stochastic volatility model for financial data which is capable of modeling long range dependence effects. The finiteness of moments and the second‐order structure of the volatility, the log‐ returns, as well as their “squares” are discussed in detail. Moreover, we give several examples in which long memory effects occur and study how the model as well as the simple Ornstein–Uhlenbeck type stochastic volatility model behave under linear transformations. In particular, the models are shown to be preserved under invertible linear transformations. Finally, we discuss how (sup)OU stochastic volatility models can be combined with a factor modeling approach.

Mehler semigroups, Ornstein-Uhlenbeck processes and background driving Lévy processes on locally compact groups and on hypergroups

For finite dimensional vector spaces it is well-known that there exists a 1-1-correspondence between distributions of Ornstein-Uhlenbeck type processes (w.r.t. a fixed group of automorphisms) and (background driving) Lévy processes, hence between M- or skew convolution semigroups on the one hand and continuous convolution semigroups on the other. An analogous result could be proved for simply connected nilpotent Lie groups. Here we extend this correspondence to a class of commutative hypergroups.

Pure -Jump Levy Processes and Self-decomposability in Financial Modeling

In this study, we review the connections between L\'{e}vy processes with jumps and self-decomposable laws. Self-decomposable laws constitute a subclass of infinitely divisible laws. L\'{e}vy processes additive processes and independent increments can be related using self-similarity property. Sato (1991) defined additive processes as a generalization of L\'{e}vy processes. In this way, additive processes are those processes with inhomogeneous (in general) and independent increments and L\'{e}vy processes correspond with the particular case in which the increments are time homogeneous. Hence L\'{e}vy processes are considerable as a particular type. Self-decomposable distributions occur as limit law an Ornstein-Uhlenbeck type process associated with a background driving L\'{e}vy process. Finally as an application, asset returns are representing by a normal inverse Gaussian process. Then to test applicability of this representation, we use the nonparametric threshold estimator of the quadratic variation, proposed by Cont andMancini (2007).

Lévy–Ornstein–Uhlenbeck transition semigroup as second quantized operator

Let μ be an invariant measure for the transition semigroup ( P t ) of the Markov family defined by the Ornstein–Uhlenbeck type equation d X = A X d t + d L on a Hilbert space E, driven by a Lévy process L. It is shown that for any t ⩾ 0 , P t considered on L 2 ( μ ) is a second quantized operator on a Poisson Fock space of e At . From this representation it follows that the transition semigroup corresponding to the equation on E = R , driven by an α-stable noise L, α ∈ ( 0 , 2 ) , is neither compact nor symmetric.

Conducting properties of classical transmission lines with Ornstein–Uhlenbeck type disorder

In this work we study the behavior of bands of extended states and localized states which appear in classical disordered electrical transmission lines, when we use a ternary map and the Ornstein–Uhlenbeck process to generate the long-range correlated disorder, instead of using the Fourier filtering method. By performing finite-size scaling we obtain the asymptotic value of the map parameter b in the thermodynamic limit in a selected range of values of the parameters γ and C of the Ornstein–Uhlenbeck process. With these data we obtain the phase diagrams which separate the localized states from the extended states. These are the fundamental results of this article.

On the Hougaard subordinated Gaussian Lévy processes

The Hougaard subordinated multivariate Gaussian Lévy processes are characterized. Necessary and sufficient conditions for their self-decomposability are given and related Ornstein–Uhlenbeck type processes are described.

A Multivariate Stochastic Hybrid Model with Switching Coefficients and Jumps: Solution and Distribution

In this work, a class of multidimensional stochastic hybrid dynamic models is studied. The system under investigation is a first‐order linear nonhomogeneous system of Itô‐Doob type stochastic differential equations with switching coefficients. The switching of the system is governed by a discrete dynamic which is monitored by a non‐homogeneous Poisson process. Closed‐form solutions of the systems are obtained. Furthermore, the major part of the work is devoted to finding closed‐form probability density functions of the solution processes of linear homogeneous and Ornstein‐Uhlenbeck type systems with jumps.

Formula omitted]-selfdecomposable distributions and related Ornstein–Uhlenbeck type processes

The concept of selfdecomposability has been generalized to that of α -selfdecomposability, α ∈ R , by many authors. We first mention the existing results on the class of α -selfdecomposable distributions and investigate the remaining problems. We give complete characterizations by stochastic integrals with respect to Lévy processes for the case 1 ≤ α < 2 . The main topic of this paper is Langevin type equations and the corresponding Ornstein–Uhlenbeck type processes related to α -selfdecomposable distributions.

Regularity of semigroups generated by Lévy type operators via coupling

By adopting the coupling method, we obtain new verifiable sufficient conditions about the C b ( R d ) -Feller continuity, the Lipschitz continuity and the strong Feller continuity of the semigroups associated with Lévy type operators. These results easily apply to jump–diffusion processes and stochastic differential equations driven by Lévy processes. Our results also yield the criterion for the e -property (namely the characterization about the equi-continuity of semigroups acting on bounded Lipschitz functions) of Lévy type operators, and show that both genuine Lévy processes and the Ornstein–Uhlenbeck type processes are e -processes.

Transition Law-based Simulation of Generalized Inverse Gaussian Ornstein–Uhlenbeck Processes

In this paper, a stochastic integral of Ornstein–Uhlenbeck type is represented to be the sum of three independent random variables—one follows a distribution whose density is a deconvolution of the densities of two generalized inverse Gaussian distributions, and the two others all have compound Poisson distributions. Based on the representation of the stochastic integral, a simulation procedure for obtaining discretely observed values of Ornstein–Uhlenbeck processes with given generalized inverse Gaussian distribution is provided. For some subclasses of the generalized inverse Gaussian Ornstein–Uhlenbeck process, the innovations can be sampled exactly. The performance of the simulation method is evidenced by some empirical results.

Simulation of multifractal products of Ornstein–Uhlenbeck type processes

This paper investigates and provides evidence of the multifractal properties of products of the exponential of Ornstein–Uhlenbeck processes driven by Lévy motion. We demonstrate in detail the construction of a multifractal process with gamma subordinator as the background driving Lévy process. Simulations are performed for the scenarios corresponding to the normal inverse Gaussian, gamma and inverse Gaussian distributions. The log periodograms and Rényi functions of the simulated processes are also computed to investigate their multifractality.

Stochastic integral characterizations of semi-selfdecomposable distributions and related Ornstein-Uhlenbeck type processes

In this paper, three topics on semi-selfdecomposable distribu- tions are studied. The flrst one is to characterize semi-selfdecomposable distributions by stochastic integrals with respect to Levy processes. This characterization deflnes a mapping from an inflnitely divisible distribution with flnite log-moment to a semi-selfdecomposable distribution. The sec- ond one is to introduce and study a Langevin type equation and the cor- responding Ornstein-Uhlenbeck type process whose limiting distribution is semi-selfdecomposable. Also, semi-stationary Ornstein-Uhlenbeck type pro- cesses with semi-selfdecomposable distributions are constructed. The third one is to study the iteration of the mapping above. The iterated mapping is expressed as a single mapping with a difierent integrand. Also, nested sub- classes of the class of semi-selfdecomposable distributions are considered, and it is shown that the limit of these nested subclasses is the closure of the class of semi-stable distributions.

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.