scholarly journals The Lévy-Itô decomposition of sample paths of Lévy processes with values in the space of probability measures

2009 ◽  
Vol 61 (1) ◽  
pp. 263-289
Author(s):  
Kouji YAMAMURO
Keyword(s):  
Lévy Processes ◽  
Levy Processes ◽  
Probability Measures ◽  
Sample Paths

Generalized fractional Lévy processes with fractional Brownian motion limit

2015 ◽  
Vol 47 (04) ◽  
pp. 1108-1131 ◽  
Author(s):  
Claudia Klüppelberg ◽  
Muneya Matsui
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Lévy Processes ◽  
Sample Path ◽  
Levy Processes ◽  
Order Structure ◽  
Uhlenbeck Process ◽  
Sample Paths ◽  
Volatility Models ◽  
Ornstein Uhlenbeck Process

Fractional Lévy processes generalize fractional Brownian motion in a natural way. We go a step further and extend the usual fractional Riemann-Liouville kernel to a regularly varying function. We call the resulting stochastic processes generalized fractional Lévy processes (GFLPs) and show that they may have short or long memory increments and that their sample paths may have jumps or not. Moreover, we define stochastic integrals with respect to a GFLP and investigate their second-order structure and sample path properties. A specific example is the Ornstein-Uhlenbeck process driven by a time-scaled GFLP. We prove a functional central limit theorem for such scaled processes with a fractional Ornstein-Uhlenbeck process as a limit process. This approximation applies to a wide class of stochastic volatility models, which include models where possibly neither the data nor the latent volatility process are semimartingales.

Sharp Localized Inequalities for Fourier Multipliers

2014 ◽  
Vol 66 (6) ◽  
pp. 1358-1381
Author(s):  
Adam Osękowski
Keyword(s):  
Lower Bounds ◽  
Lévy Processes ◽  
Levy Processes ◽  
Fourier Multipliers ◽  
Probabilistic Methods ◽  
Independent Interest ◽  
Probability Measures ◽  
Sharp Bounds ◽  
The Real

AbstractIn this paper we study sharp localized Lq → Lp estimates for Fourier multipliers resulting from modulation of the jumps of Lévy processes. The proofs of these estimates rest on probabilistic methods and exploit related sharp bounds for differentially subordinated martingales, which are of independent interest. The lower bounds for the constants involve the analysis of laminates, a family of certain special probability measures on 2×2 matrices. As an application, we obtain new sharp bounds for the real and imaginary parts of the Beurling–Ahlfors operator.

Generalized fractional Lévy processes with fractional Brownian motion limit

2015 ◽  
Vol 47 (4) ◽  
pp. 1108-1131 ◽  
Author(s):  
Claudia Klüppelberg ◽  
Muneya Matsui
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Lévy Processes ◽  
Sample Path ◽  
Levy Processes ◽  
Order Structure ◽  
Uhlenbeck Process ◽  
Sample Paths ◽  
Volatility Models ◽  
Ornstein Uhlenbeck Process

Fractional Lévy processes generalize fractional Brownian motion in a natural way. We go a step further and extend the usual fractional Riemann-Liouville kernel to a regularly varying function. We call the resulting stochastic processes generalized fractional Lévy processes (GFLPs) and show that they may have short or long memory increments and that their sample paths may have jumps or not. Moreover, we define stochastic integrals with respect to a GFLP and investigate their second-order structure and sample path properties. A specific example is the Ornstein-Uhlenbeck process driven by a time-scaled GFLP. We prove a functional central limit theorem for such scaled processes with a fractional Ornstein-Uhlenbeck process as a limit process. This approximation applies to a wide class of stochastic volatility models, which include models where possibly neither the data nor the latent volatility process are semimartingales.

scholarly journals Estimates of heat kernels of non-symmetric Lévy processes

2021 ◽  
Vol 33 (5) ◽  
pp. 1207-1236
Author(s):  
Tomasz Grzywny ◽  
Karol Szczypkowski
Keyword(s):  
Lévy Processes ◽  
Characteristic Exponent ◽  
Levy Processes ◽  
Heat Kernels ◽  
Probability Measures ◽  
Convolution Semigroups

Abstract We investigate densities of vaguely continuous convolution semigroups of probability measures on ℝ d {{\mathbb{R}^{d}}} . First, we provide results that give upper estimates in a situation when the corresponding jump measure is allowed to be highly non-symmetric. Further, we prove upper estimates of the density and its derivatives if the jump measure compares with an isotropic unimodal measure and the characteristic exponent satisfies a certain scaling condition. Lower estimates are discussed in view of a recent development in that direction, and in such a way to complement upper estimates. We apply all those results to establish precise estimates of densities of non-symmetric Lévy processes.

scholarly journals A UNIFYING APPROACH TO FRACTIONAL LÉVY PROCESSES

2013 ◽  
Vol 13 (02) ◽  
pp. 1250017 ◽  
Author(s):  
SEBASTIAN ENGELKE ◽  
JEANNETTE H. C. WOERNER
Keyword(s):  
Brownian Motion ◽  
Lévy Process ◽  
Lévy Processes ◽  
Moving Average ◽  
Levy Processes ◽  
Levy Process ◽  
Integration Theory ◽  
Sample Paths ◽  
Second Moments ◽  
Moving Average Representation

Starting from the moving average representation of fractional Brownian motion, there are two different approaches to constructing fractional Lévy processes in the literature. Applying L2-integration theory, one can keep the same moving average kernel and replace the driving Brownian motion by a pure jump Lévy process with finite second moments. Alternatively, in the framework of alpha-stable random measures, the Brownian motion is replaced by an alpha-stable Lévy process and the exponent in the kernel is reparametrized by H - 1/α. We now provide a unified approach taking kernels of the form [Formula: see text], where γ can be chosen according to the existing moments and the Blumenthal–Getoor index of the underlying Lévy process. These processes may exhibit both long and short range dependence. In addition we will examine further properties of the processes, e.g., regularity of the sample paths and the semimartingale property.

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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