Approximating ambit fields via Fourier methods

In their paper Barndorff-Nielsen et al. [4] employ so called ambit fields to model electricity spot-forward dynamics. We briefly introduce and discuss ambit fields, and introduce a novel method for approximating general ambit fields by a linear combination of ambit fields driven by exponential kernel functions (as has already been done in the null-spatial case of Lévy semistationary processes by Benth et al. [11]) by approximating the deterministic kernel function by a carefully selected finite sum. Moreover, we shall study examples, in the setting of modelling electricity forward markets, of ambit fields with singular kernel functions to illustrate the usefulness of our method for pricing purposes.     

Selfdecomposable Fields

In the present paper, we study selfdecomposability of random fields, as defined directly rather than in terms of finite-dimensional distributions. The main tools in our analysis are the master Lévy measure and the associated Lévy-Itô representation. We give the dilation criterion for selfdecomposability analogous to the classical one. Next, we give necessary and sufficient conditions (in terms of the kernel function) for a Volterra field driven by a Lévy basis to be selfdecomposable. In this context, we also study the so-called Urbanik classes of random fields. We follow this with the study of existence and selfdecomposability of integrated Volterra fields. Finally, we introduce infinitely divisible field-valued Lévy processes, give the Lévy-Itô representation associated with them and study stochastic integration with respect to such processes. We provide examples in the form of Lévy semistationary processes with a Gamma kernel and Ornstein–Uhlenbeck processes.     

Fractional generalized Lévy random fields as white noise functionals

In this paper, we construct the fractional generalized Lévy random fields (FGLRF) as tempered white noise functionals. We find that this white noise approach is very effective in investigating the properties of these fields. Under some conditions, the fractional Lévy fields in the usual sense are obtained. In addition, we also present a method to construct the anisotropic fractional generalized Lévy random fields (AFGLRF).      

Generalized fractional L��vy random fields on Gel��fand triple: A white noise approach

In this paper, under the first-order moment condition of the infinitely divisible distribution on Gel’fand triple, we use Riesz potential to construct fractional Lévy random fields on Gel’fand triple by white noise approach. We investigate the distribution and sample properties of isotropic and anisotropic fractional Lévy random fields, respectively.     

Random attractors for stochastic differential equations driven by two-sided Lévy processes

Abstract

In this paper, the asymptotic behavior of solutions for a nonlinear Marcus stochastic differential equation with multiplicative two-sided Lévy noise is studied. We plan to consider this equation as a random dynamical system. Thus, we have to interpret a Lévy noise as a two-sided metric dynamical system. For that, we have to introduce some fundamental properties of such a noise. So far most studies have only discussed two-sided Lévy processes which are defined by combining two-independent Lévy processes. In this paper, we use another definition of two-sided Lévy process by expanding the probability space. Having this metric dynamical system we will show that the Marcus stochastic differential equation with a particular drift coefficient and multiplicative noise generates a random dynamical system which has a random attractor.     

Integrability of multivariate subordinated Lévy processes in Hilbert space

We investigate multivariate subordination of Lévy processes which was first introduced by Barndorff-Nielsen et al. [O.E. Barndorff-Nielsen, F.E. Benth, and A. Veraart, Modelling electricity forward markets by ambit fields, J. Adv. Appl. Probab. (2010)], in a Hilbert space valued setting which has been introduced in Pérez-Abreu and Rocha-Arteaga [V. Pérez-Abreu and A. Rocha-Arteaga, Covariance-parameter Lévy processes in the space of trace-class operators, Infin. Dimens. Anal. Quantum Probab. Related Top. 8(1) (2005), pp. 33–54]. The processes are explicitly characterized and conditions for integrability and martingale properties are derived under various assumptions of the Lévy process and subordinator. As an application of our theory we construct explicitly some Hilbert space valued versions of Lévy processes which are popular in the univariate and multivariate case. In particular, we define a normal inverse Gaussian Lévy process in Hilbert space. The resulting process has the property that at each time all its finite dimensional projections are multivariate normal inverse Gaussian distributed as introduced in Rydberg [T. Rydberg, The normal inverse Gaussian Lévy process: Simulation and approximation, Commun. Stat. Stochastic Models 13 (1997), pp. 887–910].     

Ornstein–Uhlenbeck processes in Hilbert space with non-Gaussian stochastic volatility

We propose a non-Gaussian operator-valued extension of the Barndorff-Nielsen and Shephard stochastic volatility dynamics, defined as the square-root of an operator-valued Ornstein–Uhlenbeck process with Lévy noise and bounded drift. We derive conditions for the positive definiteness of the Ornstein–Uhlenbeck process, where in particular we must restrict to operator-valued Lévy processes with “non-decreasing paths”. It turns out that the volatility model allows for an explicit calculation of its characteristic function, showing an affine structure. We introduce another Hilbert space-valued Ornstein–Uhlenbeck process with Wiener noise perturbed by this class of stochastic volatility dynamics. Under a strong commutativity condition between the covariance operator of the Wiener process and the stochastic volatility, we can derive an analytical expression for the characteristic functional of the Ornstein–Uhlenbeck process perturbed by stochastic volatility if the noises are independent. The case of operator-valued compound Poisson processes as driving noise in the volatility is discussed as a particular example of interest. We apply our results to futures prices in commodity markets, where we discuss our proposed stochastic volatility model in light of ambit fields.     

Hyperfinite Lévy Processes

A hyperfinite Lévy process is an infinitesimal random walk (in the sense of nonstandard analysis) which with probability one is finite for all finite times. We develop the basic theory for hyperfinite Lévy processes and find a characterization in terms of transition probabilities. The standard part of a hyperfinite Lévy process is a (standard) Lévy process, and we show that given a generating triplet (γ, C, μ) for standard Lévy processes, we can construct hyperfinite Lévy processes whose standard parts correspond to this triplet. Hence all Lévy laws can be obtained from hyperfinite Lévy processes. The paper ends with a brief look at Malliavin calculus for hyperfinite Lévy processes including a version of the Clark-Haussmann-Ocone formula.     

Representations,decompositions and sample function continuity of random fields with independent increments

Standard fare in the study of representations and decompositions of processes with independent increments is pursued in the somewhat more complex setting of vector-valued random fields having independent increments over disjoint sets. Such processes are first constructed as almost surely uniformly convergent sums of Poisson type summands, that immediately yield information on sample function properties of versions. The constructions employed, which include a generalized version of the Ferguson-Klass construction with uniform convergence, are new even in the simpler setting of processes in one-dimensional time.Following these constructions, or representations, an analogue of the Lévy-Ito decomposition for Lévy processes is developed, which then enables a number of simple sample function properties of these processes to be read off from the Lévy measure in their characteristic functionals.The paper concludes with a study of general centred additive random fields and an appendix incorporating a brief survey of the theory of centred sums of independent random variables.     

Multifractal analysis of Lévy fields

We study the pointwise regularity properties of the Lévy fields introduced by T. Mori; these fields are the most natural generalization of Lévy processes to the multivariate setting. We determine their spectrum of singularities, and we show that their H?lder singularity sets satisfy a large intersection property in the sense of K. Falconer.     

Strong and Weak Disorder for Lévy Directed Polymers in Random Environment

Abstract

We consider Lévy directed polymers in the Poisson random environment. We give conditions for strong or weak disorder in terms of the Lévy exponent of symmetric Lévy process.     

On convergence of random walks generated by compound Cox processes to Lévy processes

A functional limit theorem is proved establishing weak convergence of random walks generated by compound doubly stochastic Poisson processes to Lévy processes in the Skorokhod space. As corollaries, theorems are proved on convergence of random walks with jumps having finite variances to Lévy processes with mixed normal distributions, and in particular, to stable Lévy processes.     

Singularity sets of Lévy processes

We completely describe the size and large intersection properties of the Hölder singularity sets of Lévy processes. We also study the set of times at which a given function cannot be a modulus of continuity of a Lévy process. The Hölder singularity sets of the sample paths of certain random wavelet series are investigated as well.     

Lévy Laplacians in Hida Calculus and Malliavin Calculus

Some connections between different definitions of Lévy Laplacians in the stochastic analysis are considered. Two approaches are used to define these operators. The standard one is based on the application of the theory of Sobolev–Schwartz distributions over the Wiener measure (Hida calculus). One can consider the chain of Lévy Laplacians parametrized by a real parameter with the help of this approach. One of the elements of this chain is the classical Lévy Laplacian. Another approach to defining the Lévy Laplacian is based on the application of the theory of Sobolev spaces over the Wiener measure (Malliavin calculus). It is proved that the Lévy Laplacian defined with the help of the second approach coincides with one of the elements of the chain of Lévy Laplacians, but not with the classical Lévy Laplacian, under the embedding of the Sobolev space over the Wiener measure in the space of generalized functionals over this measure. It is shown which Lévy Laplacian in the stochastic analysis is connected with the gauge fields.     

Kernel-Correlated Lévy Field Driven Forward Rate and Application to Derivative Pricing

We propose a term structure of forward rates driven by a kernel-correlated Lévy random field under the HJM framework. The kernel-correlated Lévy random field is composed of a kernel-correlated Gaussian random field and a centered Poisson random measure. We shall give a criterion to preclude arbitrage under the risk-neutral pricing measure. As applications, an interest rate derivative with general payoff functional is priced under this pricing measure.     

A new family of time-space harmonic polynomials with respect to Lévy processes

By means of a symbolic method, a new family of time-space harmonic polynomials with respect to Lévy processes is given. The coefficients of these polynomials involve a formal expression of Lévy processes by which many identities are stated. We show that this family includes classical families of polynomials such as Hermite polynomials. Poisson–Charlier polynomials result to be a linear combinations of these new polynomials, when they have the property to be time-space harmonic with respect to the compensated Poisson process. The more general class of Lévy–Sheffer polynomials is recovered as a linear combination of these new polynomials, when they are time-space harmonic with respect to Lévy processes of very general form. We show the role played by cumulants of Lévy processes, so that connections with boolean and free cumulants are also stated.     

Gaussian and sparse processes are limits of generalized Poisson processes

The theory of sparse stochastic processes offers a broad class of statistical models to study signals, far beyond the more classical class of Gaussian processes. In this framework, signals are represented as realizations of random processes that are solution of linear stochastic differential equations driven by Lévy white noises. Among these processes, generalized Poisson processes based on compound-Poisson noises admit an interpretation as random L-splines with random knots and weights. We demonstrate that every generalized Lévy process—from Gaussian to sparse—can be understood as the limit in law of a sequence of generalized Poisson processes. This enables a new conceptual understanding of sparse processes and suggests simple algorithms for the numerical generation of such objects.     

Weak approximation of the complex Brownian sheet from a Lévy sheet and applications to SPDEs

We consider a Lévy process in the plane and we use it to construct a family of complex-valued random fields that we show to converge in law, in the space of continuous functions, to a complex Brownian sheet. We apply this result to obtain weak approximations of the random field solution to a semilinear one-dimensional stochastic heat equation driven by the space–time white noise.     

Singularity among selfsimilar Gaussian random fields with different scaling parameters and others

Abstract

It is shown in this paper that the probability measures generated by selfsimilar Gaussian random fields are mutually singular, whenever they have different scaling parameters. So are those generated from a selfsimilar Gaussian random field and a stationary Gaussian random field. Certain conditions are also given for the singularity of the probability measures generated from two Gaussian random fields whose covariance functions are Schoenberg–Lévy kernels, and for those from stationary Gaussian random fields with spectral densities.     

Lévy driven CARMA generalized processes and stochastic partial differential equations

We give a new definition of a Lévy driven CARMA random field, defining it as a generalized solution of a stochastic partial differential equation (SPDE). Furthermore, we give sufficient conditions for the existence of a mild solution of our SPDE. Our model unifies all known definitions of CARMA random fields, and in particular for dimension 1 we obtain the classical CARMA process.