scholarly journals Lévy processes and Lévy white noise as tempered distributions

2017 ◽  
Vol 45 (6B) ◽  
pp. 4389-4418 ◽  
Author(s):  
Robert C. Dalang ◽  
Thomas Humeau
Keyword(s):  
White Noise ◽  
Lévy Processes ◽  
Levy Processes ◽  
Tempered Distributions ◽  
Lévy White Noise

GENERALIZED FRACTIONAL LÉVY PROCESSES: A WHITE NOISE APPROACH

2006 ◽  
Vol 06 (04) ◽  
pp. 473-485 ◽  
Author(s):  
ZHI YUAN HUANG ◽  
PEI YAN LI
Keyword(s):  
White Noise ◽  
Lévy Processes ◽  
Sufficient Conditions ◽  
Levy Processes ◽  
Simple Condition ◽  
Lévy Measure ◽  
Lévy White Noise ◽  
White Noise Functionals

In this paper, with a simple condition on Lévy measure, we construct the (tempered) generalized fractional Lévy processes (GFLP) as Lévy white noise functionals and investigate their distribution and sample properties through this white noise approach. We also give some sufficient conditions under which the usual fractional Lévy processes (FLP) are well defined.

scholarly journals White noise analysis for Lévy processes

2004 ◽  
Vol 206 (1) ◽  
pp. 109-148 ◽  
Author(s):  
Giulia Di Nunno ◽  
Bernt Øksendal ◽  
Frank Proske
Keyword(s):  
White Noise ◽  
Lévy Processes ◽  
Noise Analysis ◽  
Levy Processes ◽  
White Noise Analysis

scholarly journals Infinite dimensional analysis of pure jump Lévy processes on the Poisson space

2006 ◽  
Vol 98 (2) ◽  
pp. 237 ◽  
Author(s):  
Arne Løkka ◽  
Frank Norbert Proske
Keyword(s):  
White Noise ◽  
Lévy Processes ◽  
Levy Processes ◽  
Wick Product ◽  
Charlier Polynomials ◽  
Starting Point ◽  
Poisson Space ◽  
Lévy Measures ◽  
Distribution Spaces ◽  
White Noise Calculus

We develop a white noise calculus for pure jump Lévy processes on the Poisson space. This theory covers the treatment of Lévy processes of unbounded variation. The starting point of the theory is the construction of a distribution space. This space has many of the same nice properties as the classical Schwartz space, but is modified in a certain way in order to be more suitable for pure jump Lévy processes. We apply Minlos's theorem to this space and obtain a white noise measure which satisfies the first condition of analyticity, and which is non-degenerate. Furthermore, we obtain generalized Charlier polynomials for all Lévy measures. We introduce Kondratiev test function and distribution spaces, the $\mathcal{S}$-transform and the Wick product. We proceed by using a transfer principle on Poisson spaces to establish a differential calculus.

AN APPLICATION OF THE SEGAL–BARGMANN TRANSFORM TO THE CHARACTERIZATION OF LÉVY WHITE NOISE MEASURES

2010 ◽  
Vol 13 (02) ◽  
pp. 191-221 ◽  
Author(s):  
YUH-JIA LEE ◽  
HSIN-HUNG SHIH
Keyword(s):  
White Noise ◽  
Probability Measures ◽  
Infinitely Divisible Distributions ◽  
Infinite Dimensions ◽  
Infinitely Divisible ◽  
Tempered Distributions ◽  
Bargmann Transform ◽  
Second Moment ◽  
Lévy White Noise

Being inspired by the observation that the Stein's identity is closely connected to the quantum decomposition of probability measures and the Segal–Bargmann transform, we are able to characterize the Lévy white noise measures on the space [Formula: see text] of tempered distributions associated with a Lévy spectrum having finite second moment. The results not only extends the Stein and Chen's lemma for Gaussian and Poisson distributions to infinite dimensions but also to many other infinitely divisible distributions such as Gamma and Pascal distributions and corresponding Lévy white noise measures on [Formula: see text].

On correlating Lévy processes

2010 ◽  
Vol 13 (1) ◽  
pp. 3-16 ◽  
Author(s):  
Ernst Eberlein ◽  
Dilip Madan
Keyword(s):  
Lévy Processes ◽  
Levy Processes
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