Integrability of multivariate subordinated Lévy processes in Hilbert space
Fred Espen Benth,Paul Krühner +1 more
TLDR
In this paper, a normal inverse Gaussian Levy process in Hilbert space is defined and conditions for integrability and martingale properties are derived under various assumptions of the Levy process and subordinator.Abstract:
We investigate multivariate subordination of Levy 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 Perez-Abreu and Rocha-Arteaga [V. Perez-Abreu and A. Rocha-Arteaga, Covariance-parameter Levy 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 Levy process and subordinator. As an application of our theory we construct explicitly some Hilbert space valued versions of Levy processes which are popular in the univariate and multivariate case. In particular, we define a normal inverse Gaussian Levy process in Hilbert space. The resulting process has the property that at each time ...read more
Citations
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Lévy processes and infinitely divisible distributions
TL;DR: In this paper, the authors consider the distributional properties of Levy processes and propose a potential theory for Levy processes, which is based on the Wiener-Hopf factorization.
Journal ArticleDOI
Real and Complex Analysis. By W. Rudin. Pp. 412. 84s. 1966. (McGraw-Hill, New York.)
TL;DR: In this paper, the Riesz representation theorem is used to describe the regularity properties of Borel measures and their relation to the Radon-Nikodym theorem of continuous functions.
Journal ArticleDOI
Approximation and simulation of infinite-dimensional Lévy processes
Andrea Barth,Andreas Stein +1 more
TL;DR: In this paper, the authors introduced approximation methods for infinite-dimensional Levy processes, also called (time-dependent) Levy fields, where the point-wise marginal distributions are dependent but uncorrelated subordinated Wiener processes.
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Approximation and simulation of infinite-dimensional Levy processes
Andrea Barth,Andreas Stein +1 more
TL;DR: In this paper, the authors introduced approximation methods for infinite-dimensional Levy processes, also called (time-dependent) Levy fields, where the point-wise marginal distributions are dependent but uncorrelated subordinated Wiener processes.
Journal ArticleDOI
On stochastic control for time changed Lévy dynamics
TL;DR: In this article , an approach to stochastic control via maximum principle for time changed Lévy dynamics is presented, which can be extended to Volterra type dynamics and the control of forward-backward systems of equations.
References
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Limit Theorems for Stochastic Processes
Jean Jacod,Albert N. Shiryaev +1 more
TL;DR: In this article, the General Theory of Stochastic Processes, Semimartingales, and Stochastically Integrals is discussed and the convergence of Processes with Independent Increments is discussed.
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The Variance Gamma Process and Option Pricing
TL;DR: In this article, a three-parameter stochastic process, termed the variance gamma process, is developed as a model for the dynamics of log stock prices, which is obtained by evaluating Brownian motion with drift at a random time given by a gamma process.
Journal ArticleDOI
The Variance Gamma (V.G.) Model for Share Market Returns
Dilip B. Madan,Eugene Seneta +1 more
TL;DR: In this paper, a new stochastic process, termed the variance gamma process, is proposed as a model for the uncertainty underlying security prices, which is normal conditional on a variance, distributed as a gamma variate.
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Probability measures in a metric space
TL;DR: In this article, the authors provide an overview on probability measures in a metric space and present a smaller class of measures on metric spaces called tight measures, which have the property that they are determined by their values for compact sets.
Journal ArticleDOI
The normal inverse gaussian lévy process: simulation and approximation
TL;DR: In this article, the one and two-dimensional normal inverse Gaussian Levy process is studied in relation to German and Danish financial data and the uniform residuals are calculated by means of an algorithm which simulates random variables from the normal inverse GAussian distribution.