Approximation for the solutions of stochastic differential equations: ii strong convergence

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.

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Strong convergence and asymptotic stability of explicit numerical schemes for nonlinear stochastic differential equations

Mathematics of Computation ◽

10.1090/mcom/3661 ◽

2021 ◽

pp. 1

Author(s):

Xiaoyue Li ◽

Xuerong Mao ◽

Hongfu Yang

Keyword(s):

Asymptotic Stability ◽

Differential Equations ◽

Strong Convergence ◽

Stochastic Differential Equations ◽

Numerical Schemes ◽

Nonlinear Stochastic Differential Equations

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

Monte Carlo Methods and Applications ◽

10.1515/mcma-2020-2057 ◽

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.

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Optimal rate of convergence for two classes of schemes to stochastic differential equations driven by fractional Brownian motions

IMA Journal of Numerical Analysis ◽

10.1093/imanum/draa019 ◽

2020 ◽

Author(s):

Jialin Hong ◽

Chuying Huang ◽

Xu Wang

Keyword(s):

Differential Equations ◽

Convergence Rate ◽

Strong Convergence ◽

Stochastic Differential Equations ◽

Numerical Solutions ◽

Brownian Motions ◽

Fractional Brownian Motions ◽

Optimal Rate Of Convergence ◽

Runge Kutta ◽

Strong Convergence Rate

Abstract This paper investigates numerical schemes for stochastic differential equations driven by multi-dimensional fractional Brownian motions (fBms) with Hurst parameter $H\in (\frac 12,1)$. Based on the continuous dependence of numerical solutions on the driving noises, we propose the order conditions of Runge–Kutta methods for the strong convergence rate $2H-\frac 12$, which is the optimal strong convergence rate for approximating the Lévy area of fBms. We provide an alternative way to analyse the convergence rate of explicit schemes by adding ‘stage values’ such that the schemes are interpreted as Runge–Kutta methods. Taking advantage of this technique the strong convergence rate of simplified step-$N$ Euler schemes is obtained, which gives an answer to a conjecture in Deya et al. (2012) when $H\in (\frac 12,1)$. Numerical experiments verify the theoretical convergence rate.

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