Dynamic models of long-memory processes driven by Lévy noise

2002 ◽  
Vol 39 (4) ◽  
pp. 730-747 ◽  
Author(s):  
V. V. Anh ◽  
C. C. Heyde ◽  
N. N. Leonenko
Keyword(s):  
Continuous Time ◽  
Long Memory ◽  
Heavy Tails ◽  
Dynamic Models ◽  
Lévy Noise ◽  
Long Memory Processes ◽  
Fractional Differential ◽  
The Green Function ◽  
Levy Noise ◽  
Modelling Data

A class of continuous-time models is developed for modelling data with heavy tails and long-range dependence. These models are based on the Green function solutions of fractional differential equations driven by Lévy noise. Some exact results on the second- and higher-order characteristics of the equations are obtained. Applications to stochastic volatility of asset prices and macroeconomics are provided.

Dynamic models of long-memory processes driven by Lévy noise

2002 ◽  
Vol 39 (04) ◽  
pp. 730-747 ◽  
Author(s):  
V. V. Anh ◽  
C. C. Heyde ◽  
N. N. Leonenko
Keyword(s):  
Continuous Time ◽  
Long Memory ◽  
Heavy Tails ◽  
Dynamic Models ◽  
Lévy Noise ◽  
Long Memory Processes ◽  
Fractional Differential ◽  
The Green Function ◽  
Levy Noise ◽  
Modelling Data

A class of continuous-time models is developed for modelling data with heavy tails and long-range dependence. These models are based on the Green function solutions of fractional differential equations driven by Lévy noise. Some exact results on the second- and higher-order characteristics of the equations are obtained. Applications to stochastic volatility of asset prices and macroeconomics are provided.

An averaging principle for neutral stochastic fractional order differential equations with variable delays driven by Lévy noise

2021 ◽  
Author(s):  
Guangjun Shen ◽  
Jiang-Lun Wu ◽  
Ruidong Xiao ◽  
Xiuwei Yin
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Lévy Noise ◽  
Averaging Principle ◽  
Fractional Order Differential Equations ◽  
Variable Delays ◽  
Fractional Differential ◽  
Stochastic Fractional Differential Equations ◽  
Levy Noise ◽  
Convergence In Mean

In this paper, we establish an averaging principle for neutral stochastic fractional differential equations with non-Lipschitz coefficients and with variable delays, driven by Lévy noise. Our result shows that the solutions of the equations concerned can be approximated by the solutions of averaged neutral stochastic fractional differential equations in the sense of convergence in mean square. As an application, we present an example with numerical simulations to explore the established averaging principle.

scholarly journals Fractional differential equations driven by Lévy noise

2003 ◽  
Vol 16 (2) ◽  
pp. 97-119 ◽  
Author(s):  
V. V. Anh ◽  
R. McVinish
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Numerical Scheme ◽  
Lévy Noise ◽  
Singularity Spectrum ◽  
Prediction Formula ◽  
Sample Paths ◽  
Fractional Differential ◽  
Levy Noise ◽  
Convergence Of The Scheme

This paper considers a general class of fractional differential equations driven by Lévy noise. The singularity spectrum for these equations is obtained. This result allows to determine the conditions under which the solution is a semimartingale. The prediction formula and a numerical scheme for approximating the sample paths of these equations are given. Almost sure, uniform convergence of the scheme and some numerical results are also provided.

scholarly journals FIRST EXIT TIMES OF SOLUTIONS OF STOCHASTIC DIFFERENTIAL EQUATIONS DRIVEN BY MULTIPLICATIVE LÉVY NOISE WITH HEAVY TAILS

2011 ◽  
Vol 11 (02n03) ◽  
pp. 495-519 ◽  
Author(s):  
ILYA PAVLYUKEVICH
Keyword(s):  
Dynamical System ◽  
Differential Equations ◽  
Heavy Tails ◽  
Multiplicative Noise ◽  
Exit Time ◽  
Lévy Noise ◽  
Exit Times ◽  
First Exit Time ◽  
First Exit Times ◽  
Levy Noise

In this paper, we study first exit times from a bounded domain of a gradient dynamical system Ẏt = -∇U(Yt) perturbed by a small multiplicative Lévy noise with heavy tails. A special attention is paid to the way the multiplicative noise is introduced. In particular, we determine the asymptotics of the first exit time of solutions of Itô, Stratonovich and Marcus canonical SDEs.

scholarly journals Stabilization for hybrid stochastic differential equations driven by Lévy noise via periodically intermittent control

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Yong Ren ◽  
Qi Zhang
Keyword(s):  
Continuous Time ◽  
Stochastic Systems ◽  
Main Idea ◽  
Lévy Noise ◽  
Intermittent Control ◽  
Controlled System ◽  
Hybrid Stochastic Systems ◽  
Levy Noise ◽  
Stability Results ◽  
Drift Coefficient

<p style='text-indent:20px;'>In this work, the issue of stabilization for a class of continuous-time hybrid stochastic systems with Lévy noise (HLSDEs, in short) is explored by using periodic intermittent control. As for the unstable HLSDEs, we design a periodic intermittent controller. The main idea is to compare the controlled system with a stabilized one with a periodic intermittent drift coefficient, which enables us to use the existing stability results on the HLSDEs. An illustrative example is proposed to show the feasibility of the obtained result.</p>

Stochastic resonance with multiplicative heavy-tailed Lévy noise: Optimal tuning on an algebraic time scale

2017 ◽  
Vol 17 (04) ◽  
pp. 1750027 ◽  
Author(s):  
Isabelle Kuhwald ◽  
Ilya Pavlyukevich
Keyword(s):  
Stochastic Resonance ◽  
Heavy Tails ◽  
Bistable System ◽  
Lévy Noise ◽  
Small Noise ◽  
Optimal Tuning ◽  
Noise Perturbation ◽  
Heavy Tailed ◽  
Synchronization Effect ◽  
Levy Noise

Stochastic resonance is an amplification and synchronization effect of weak periodic signals in nonlinear systems through a small noise perturbation. In this paper we study the dynamics of stochastic resonance in a bistable system driven by multiplicative Lévy noise with heavy tails, e.g., [Formula: see text]-stable Lévy noise. We determine the optimal tuning with respect to a probabilistic synchronization measure for both the jump-diffusion and the reduced two-state Markov chain. These results extend the theory of stochastic resonance to the case of heavy tail Lévy perturbations.

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