scholarly journals On Stochastic Differential Equations Generating Non-gaussian Continuous Markov Process

2021 ◽  
Vol 17 ◽  
pp. 65-68
Author(s):  
Vladimir Lyandres
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Diffusion Coefficients ◽  
Function Estimation ◽  
Stationary Probability Distribution ◽  
Nonlinear Stochastic Differential Equations ◽  
Continuous Markov Process ◽  
Non Gaussian ◽  
The Given ◽  
And Diffusion

Continuous Markov processes widely used as a tool for modeling random phenomena in numerous applications, can be defined as solutions of generally nonlinear stochastic differential equations (SDEs) with certain drift and diffusion coefficients which together governs the process’ probability density and correlation functions. Usually it is assumed that the diffusion coefficient does not depend on the process' current value. For presentation of non-Gaussian real processes this assumption becomes undesirable, leads generally to complexity of the correlation function estimation. We consider its analysis for the process with particular pairs of the drift and diffusion coefficients providing the given stationary probability distribution of the considered process

scholarly journals On the Correlation Function of an Arbitrary Distributed Continuous Markov Process

2020 ◽  
Vol 19 ◽  
Keyword(s):  
Correlation Function ◽  
Diffusion Coefficients ◽  
Function Estimation ◽  
Stationary Probability Distribution ◽  
Nonlinear Stochastic Differential Equations ◽  
Continuous Markov Process ◽  
Current Value ◽  
Non Gaussian ◽  
The Given ◽  
And Diffusion

Continuous Markov processes widely used as a tool for modeling random phenomena in numerous applications, can be defined as solutions of generally nonlinear stochastic differential equations (SDEs) with certain drift and diffusion coefficients which together governs the process’ probability density and correlation functions. Usually it is assumed that the diffusion coefficient does not depend on the process' current value. Sometimes, in particular for presentation of non- Gaussian real processes this assumption becomes undesirable, leads generally to complexity of the correlation function estimation. We consider its analysis for the process with arbitrary pair of the drift and diffusion coefficients providing the given stationary probability distribution of the considered process.

Approximation of Euler–Maruyama for one-dimensional stochastic differential equations involving the maximum process

2020 ◽  
Vol 26 (1) ◽  
pp. 33-47
Author(s):  
Kamal Hiderah
Keyword(s):  
Differential Equations ◽  
Strong Convergence ◽  
Stochastic Differential Equations ◽  
Diffusion Coefficients ◽  
One Dimensional ◽  
Maximum Process ◽  
And Diffusion

AbstractThe aim of this paper is to show the approximation of Euler–Maruyama {X_{t}^{n}} for one-dimensional stochastic differential equations involving the maximum process. In addition to that it proves the strong convergence of the Euler–Maruyama whose both drift and diffusion coefficients are Lipschitz. After that, it generalizes to the non-Lipschitz case.

Strong rate of convergence for the Euler–Maruyama approximation of one-dimensional stochastic differential equations involving the local time at point zero

2018 ◽  
Vol 24 (4) ◽  
pp. 249-262
Author(s):  
Mohsine Benabdallah ◽  
Kamal Hiderah
Keyword(s):  
Differential Equations ◽  
Strong Convergence ◽  
Stochastic Differential Equations ◽  
Rate Of Convergence ◽  
Local Time ◽  
Diffusion Coefficients ◽  
One Dimensional ◽  
And Diffusion

Abstract We present the Euler–Maruyama approximation for one-dimensional stochastic differential equations involving the local time at point zero. Also, we prove the strong convergence of the Euler–Maruyama approximation whose both drift and diffusion coefficients are Lipschitz. After that, we generalize to the non-Lipschitz case.

scholarly journals Existence and Uniqueness of Solutions for a Class of Nonlinear Stochastic Differential Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-7
Author(s):  
Iryna Volodymyrivna Komashynska
Keyword(s):  
Differential Equations ◽  
Diffusion Process ◽  
Stochastic Differential Equations ◽  
Successive Approximation ◽  
Diffusion Coefficients ◽  
Existence And Uniqueness ◽  
Uniqueness Result ◽  
Uniqueness Of Solutions ◽  
Nonlinear Stochastic Differential Equations

By using successive approximation, we prove existence and uniqueness result for a class of nonlinear stochastic differential equations. Moreover, it is shown that the solution of such equations is a diffusion process and its diffusion coefficients are found.

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.

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