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.

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 Convergence Analysis of Split-Step θ-Scheme for Nonlinear Stochastic Differential Equations with Jumps

2016 ◽  
Vol 8 (6) ◽  
pp. 1004-1022 ◽  
Author(s):  
Xu Yang ◽  
Weidong Zhao
Keyword(s):  
Differential Equations ◽  
Theoretical Result ◽  
Strong Convergence ◽  
Stochastic Differential Equations ◽  
Rate Of Convergence ◽  
Numerical Experiments ◽  
Mean Square ◽  
Nonlinear Stochastic Differential Equations ◽  
Mean Square Convergence ◽  
The Mean

AbstractIn this paper, we investigate the mean-square convergence of the split-step θ-scheme for nonlinear stochastic differential equations with jumps. Under some standard assumptions, we rigorously prove that the strong rate of convergence of the split-step θ-scheme in strong sense is one half. Some numerical experiments are carried out to assert our theoretical result.

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

Approximation of Euler–Maruyama for one-dimensional stochastic differential equations involving the local times of the unknown process

2016 ◽  
Vol 22 (4) ◽  
Author(s):  
Mohsine Benabdallah ◽  
Youssfi Elkettani ◽  
Kamal Hiderah
Keyword(s):  
Weak Convergence ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Local Time ◽  
Local Times ◽  
One Dimensional ◽  
Strong And Weak Convergence

AbstractIn this paper, we consider both, the strong and weak convergence of the Euler–Maruyama approximation for one-dimensional stochastic differential equations involving the local times of the unknown process. We use a transformation in order to remove the local timeHere

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