Bayesian Inference for Continuous-Time Arma Models Driven by Non-Gaussian LÉVY Processes

2006 ◽  
Author(s):  
S.J. Godsill ◽  
G. Yang
Keyword(s):  
Bayesian Inference ◽  
Continuous Time ◽  
Lévy Processes ◽  
Levy Processes ◽  
Arma Models ◽  
Non Gaussian

Governing equations for probability densities of Marcus stochastic differential equations with Lévy noise

2016 ◽  
Vol 17 (05) ◽  
pp. 1750033 ◽  
Author(s):  
Xu Sun ◽  
Xiaofan Li ◽  
Yayun Zheng
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Probability Density ◽  
Lévy Processes ◽  
Planck Equation ◽  
Levy Processes ◽  
Stochastic Dynamical Systems ◽  
Non Gaussian ◽  
Fokker Planck Equations ◽  
Fokker Planck

Marcus stochastic differential equations (SDEs) often are appropriate models for stochastic dynamical systems driven by non-Gaussian Lévy processes and have wide applications in engineering and physical sciences. The probability density of the solution to an SDE offers complete statistical information on the underlying stochastic process. Explicit formula for the Fokker–Planck equation, the governing equation for the probability density, is well-known when the SDE is driven by a Brownian motion. In this paper, we address the open question of finding the Fokker–Planck equations for Marcus SDEs in arbitrary dimensions driven by non-Gaussian Lévy processes. The equations are given in a simple form that facilitates theoretical analysis and numerical computation. Several examples are presented to illustrate how the theoretical results can be applied to obtain Fokker–Planck equations for Marcus SDEs driven by Lévy processes.

scholarly journals TOPOLOGICAL EQUIVALENCE FOR DISCONTINUOUS RANDOM DYNAMICAL SYSTEMS AND APPLICATIONS

2013 ◽  
Vol 14 (01) ◽  
pp. 1350007 ◽  
Author(s):  
HUIJIE QIAO ◽  
JINQIAO DUAN
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Stochastic Differential Equation ◽  
Stochastic Differential Equations ◽  
Lévy Processes ◽  
Levy Processes ◽  
Topological Equivalence ◽  
Random Differential Equation ◽  
Jump Discontinuities ◽  
Non Gaussian

After defining non-Gaussian Lévy processes for two-sided time, stochastic differential equations with such Lévy processes are considered. Solution paths for these stochastic differential equations have countable jump discontinuities in time. Topological equivalence (or conjugacy) for such an Itô stochastic differential equation and its transformed random differential equation is established. Consequently, a stochastic Hartman–Grobman theorem is proved for the linearization of the Itô stochastic differential equation. Furthermore, for Marcus stochastic differential equations, this topological equivalence is used to prove the existence of global random attractors.

scholarly journals TIME SERIES REGRESSION ON INTEGRATED CONTINUOUS-TIME PROCESSES WITH HEAVY AND LIGHT TAILS

2012 ◽  
Vol 29 (1) ◽  
pp. 28-67 ◽  
Author(s):  
Vicky Fasen
Keyword(s):  
Continuous Time ◽  
Lévy Processes ◽  
Levy Processes ◽  
Multiple Regression Model ◽  
Time Series Regression ◽  
Light Tails ◽  
Additional Noise ◽  
Asymptotic Confidence Intervals ◽  
Continuous Time Processes ◽  
Integrated Processes

The paper presents a cointegration model in continuous time, where the linear combinations of the integrated processes are modeled by a multivariate Ornstein–Uhlenbeck process. The integrated processes are defined as vector-valued Lévy processes with an additional noise term. Hence, if we observe the process at discrete time points, we obtain a multiple regression model. As an estimator for the regression parameter we use the least squares estimator. We show that it is a consistent estimator and derive its asymptotic behavior. The limit distribution is a ratio of functionals of Brownian motions and stable Lévy processes, whose characteristic triplets have an explicit analytic representation. In particular, we present the Wald and the t-ratio statistic and simulate asymptotic confidence intervals. For the proofs we derive some central limit theorems for multivariate Ornstein–Uhlenbeck processes.

scholarly journals Amalgamated Free Lévy Processes as Limits of Sample Covariance Matrices

2022 ◽  
Author(s):  
G. L. Zitelli
Keyword(s):  
Lévy Processes ◽  
Matrix Theory ◽  
Free Probability ◽  
Levy Processes ◽  
Covariance Matrices ◽  
Sample Covariance Matrices ◽  
Sample Covariance ◽  
Probability Spaces ◽  
Spectral Distributions ◽  
Non Gaussian

AbstractWe prove the existence of joint limiting spectral distributions for families of random sample covariance matrices modeled on fluctuations of discretized Lévy processes. These models were first considered in applications of random matrix theory to financial data, where datasets exhibit both strong multicollinearity and non-normality. When the underlying Lévy process is non-Gaussian, we show that the limiting spectral distributions are distinct from Marčenko–Pastur. In the context of operator-valued free probability, it is shown that the algebras generated by these families are asymptotically free with amalgamation over the diagonal subalgebra. This framework is used to construct operator-valued $$^*$$ ∗ -probability spaces, where the limits of sample covariance matrices play the role of non-commutative Lévy processes whose increments are free with amalgamation.

scholarly journals Low-frequency estimation of continuous-time moving average Lévy processes

Bernoulli ◽  
2019 ◽  
Vol 25 (2) ◽  
pp. 902-931 ◽  
Author(s):  
Denis Belomestny ◽  
Vladimir Panov ◽  
Jeannette H.C. Woerner
Keyword(s):  
Continuous Time ◽  
Lévy Processes ◽  
Frequency Estimation ◽  
Moving Average ◽  
Low Frequency ◽  
Levy Processes

scholarly journals Methods of simulation mathematical modeling of the Russian derivatives market in modern times

2020 ◽  
Vol 19 (4) ◽  
pp. 398-406
Author(s):  
T. A. Karpinskaya ◽  
O. E. Kudryavtsev
Keyword(s):  
Mathematical Modeling ◽  
Lévy Processes ◽  
Option Price ◽  
Levy Processes ◽  
Software Implementation ◽  
Application Development ◽  
Mathematical Methods ◽  
Development Environment ◽  
Generalized Poisson ◽  
Non Gaussian

Introduction. The paper is devoted to simulation modeling. Basic methods of the simulation mathematical modeling in the derivatives market are described. A group of realistic nonGaussian Levy processes that generalize the classical BlackScholes model is considered. The work objective is to study the most efficient methods of market forecasting, as well as the software implementation of the simulation mathematical modeling technique of the Russian derivatives market based on the Levy model. This research is relevant due to the demand for applications that simulate the dynamics of financial assets and evaluate options in realistic models of the derivatives market, allowing for jumps.Materials and Methods. Basic methods for forecasting the derivatives market, methods for determining the volatility rate at a known option price, are considered. The most effective types of Levy processes for the simulation mathematical modeling of the Russian derivatives market at the present stage are highlighted. The possibilities of the Java language for the implementation of mathematical methods are considered.Research Results. A program is developed in the Java programming language that implements the Levy mathematical model, which includes Gaussian and generalized Poisson processes. The program for calculating the mathematical method is created in the free integrated application development environment NetBeans IDE to work with any operating system.Discussion and Conclusions. The result of the simulation mathematical modeling analysis has shown that the most efficient methods in the derivatives market are those based on realistic non-Gaussian Levy processes. The software implementation of such mathematical methods can be used for educational purposes. The developed application has demonstrated high quality and speed of calculations using software resources.

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