scholarly journals The truncated Milstein method for super-linear stochastic differential equations with Markovian switching

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Weijun Zhan ◽  
Qian Guo ◽  
Yuhao Cong
Keyword(s):  
Markov Chain ◽  
Differential Equations ◽  
Convergence Rate ◽  
Stochastic Differential Equations ◽  
Convergence Result ◽  
Convergence Order ◽  
Mean Square ◽  
Numerical Example ◽  
Milstein Method ◽  
The Mean

<p style='text-indent:20px;'>In this paper, to approximate the super-linear stochastic differential equations modulated by a Markov chain, we investigate a truncated Milstein method with convergence order 1 in the mean-square sense. Under Khasminskii-type conditions, we establish the convergence result by employing a relationship between local and global errors. Finally, we confirm the convergence rate by a numerical example.</p>

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 A Compensated Numerical Method for Solving Stochastic Differential Equations with Variable Delays and Random Jump Magnitudes

2015 ◽  
Vol 2015 ◽  
pp. 1-11
Author(s):  
Ying Du ◽  
Changlin Mei
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Numerical Method ◽  
Analytical Solutions ◽  
Numerical Solutions ◽  
Mean Square ◽  
Variable Delays ◽  
Mean Square Stability ◽  
The Mean ◽  
Random Jump

Stochastic differential equations with jumps are of a wide application area especially in mathematical finance. In general, it is hard to obtain their analytical solutions and the construction of some numerical solutions with good performance is therefore an important task in practice. In this study, a compensated split-stepθmethod is proposed to numerically solve the stochastic differential equations with variable delays and random jump magnitudes. It is proved that the numerical solutions converge to the analytical solutions in mean-square with the approximate rate of 1/2. Furthermore, the mean-square stability of the exact solutions and the numerical solutions are investigated via a linear test equation and the results show that the proposed numerical method shares both the mean-square stability and the so-called A-stability.

scholarly journals Numerical Spline Method for Simulation of Stochastic Differential Equations systems: طريقة شرائحية عددية لمحاكاة حل نظم من المعادلات التفاضلية العشوائية

2021 ◽  
Vol 5 (4) ◽  
pp. 130-111
Author(s):  
Suliman M. Mahmoud, Ahmad Al-Wassouf, Ali S. Ehsaan Suliman M. Mahmoud, Ahmad Al-Wassouf, Ali S. Ehsaan
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Continuous Process ◽  
Mean Square ◽  
Spline Method ◽  
Third Order ◽  
Mean Square Stability ◽  
The Mean ◽  
Linear Problems ◽  
Two Parameters

In this paper, numerical spline method is presented with collocation two parameters for solving systems of multi-dimensional stochastic differential equations (SDEs). Multi-Wiener's time-continuous process is simulated as a discrete process, and then the mean-square stability of proposed method when applied to a system of two-dimensional linear SDEs is studied. The study shows that the method is mean-square stability and third-order convergent when applied to a system of linear and nonlinear SDEs. Moreover, the effectiveness of our method was tested by solving two test linear and non-linear problems. The numerical results show that the accuracy and applicability of the proposed method are worthy of attention.

scholarly journals Mean-Square Stability of Split-Step Theta Milstein Methods for Stochastic Differential Equations

2018 ◽  
Vol 2018 ◽  
pp. 1-13 ◽  
Author(s):  
Mahmoud A. Eissa ◽  
Haiying Zhang ◽  
Yu Xiao
Keyword(s):  
Numerical Methods ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Mean Square ◽  
Step Size ◽  
Mean Square Stability ◽  
Test Equation ◽  
The Mean ◽  
The Stability ◽  
And Diffusion

The fundamental analysis of numerical methods for stochastic differential equations (SDEs) has been improved by constructing new split-step numerical methods. In this paper, we are interested in studying the mean-square (MS) stability of the new general drifting split-step theta Milstein (DSSθM) methods for SDEs. First, we consider scalar linear SDEs. The stability function of the DSSθM methods is investigated. Furthermore, the stability regions of the DSSθM methods are compared with those of test equation, and it is proved that the methods with θ≥3/2 are stochastically A-stable. Second, the nonlinear stability of DSSθM methods is studied. Under a coupled condition on the drifting and diffusion coefficients, it is proved that the methods with θ>1/2 can preserve the MS stability of the SDEs with no restriction on the step-size. Finally, numerical examples are given to examine the accuracy of the proposed methods under the stability conditions in approximation of SDEs.

scholarly journals Stability of a Class of Nonlinear Neutral Stochastic Differential Equations with Variable Time Delays

2012 ◽  
Vol 20 (1) ◽  
pp. 467-488 ◽  
Author(s):  
Meng Wu ◽  
Nanjing Huang ◽  
Changwen Zhao
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Time Delays ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Mean Square ◽  
Variable Time ◽  
Mean Square Stability ◽  
The Mean ◽  
Necessary And Sufficient

Abstract In this paper, we study the mean square asymptotic stability of a class of generalized nonlinear neutral stochastic differential equations with variable time delays by using fixed point theory. An asymptotic mean square stability theorem with a necessary and sufficient condition is proved which improves and generalizes some well-known results. Finally, two examples are given to illustrate our results.

scholarly journals On the averaging principle for SDEs driven by G-Brownian motion with non-Lipschitz coefficients

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Wei Mao ◽  
Bo Chen ◽  
Surong You
Keyword(s):  
Brownian Motion ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Averaging Principle ◽  
Mean Square ◽  
The Mean

AbstractIn this paper, we aim to develop the averaging principle for stochastic differential equations driven by G-Brownian motion (G-SDEs for short) with non-Lipschitz coefficients. By the properties of G-Brownian motion and stochastic inequality, we prove that the solution of the averaged G-SDEs converges to that of the standard one in the mean-square sense and also in capacity. Finally, two examples are presented to illustrate our theory.

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