Convergence and Stability of the Truncated Euler-Maruyama Method for Stochastic Differential Equations with Piecewise Continuous Arguments

In this paper, we develop the truncated Euler-Maruyama (EM) method for stochastic differential equations with piecewise continuous arguments (SDEPCAs), and consider the strong convergence theory under the local Lipschitz condition plus the Khasminskii-type condition. The order of convergence is obtained. Moreover, we show that the truncated EM method can preserve the exponential mean square stability of SDEPCAs. Numerical examples are provided to support our conclusions.     

Strong Convergence of the Euler-Maruyama Method for a Class of Stochastic Volterra Integral Equations

In this paper, we consider the Euler-Maruyama method for a class of stochastic Volterra integral equations (SVIEs). It is known that the strong convergence order of the Euler-Maruyama method is $\frac12$. However, the strong superconvergence order $1$ can be obtained for a class of SVIEs if the kernels $\sigma_{i}(t, t) = 0$ for $i=1$ and $2$; otherwise, the strong convergence order is $\frac12$. Moreover, the theoretical results are illustrated by some numerical examples.     

The Convergence of Truncated Euler-Maruyama Method for Stochastic Differential Equations with Piecewise Continuous Arguments Under Generalized One-Sided Lipschitz Condition

In this paper, we consider the stochastic differential equations with piecewise continuous arguments (SDEPCAs) in which the drift coefficient satisfies the generalized one-sided Lipschitz condition and the diffusion coefficient satisfies the linear growth condition. Since the delay term $t-[t]$ of SDEPCAs is not continuous and differentiable, the variable substitution method is not suitable. To overcome this difficulty, we adopt new techniques to prove the boundedness of the exact solution and the numerical solution. It is proved that the truncated Euler-Maruyama method is strongly convergent to SDEPCAs in the sense of $L^{\bar{q}}(\bar{q}\ge 2)$. We obtain the convergence order with some additional conditions. An example is presented to illustrate the analytical theory.     

Strong convergence rate of truncated Euler-Maruyama method for stochastic differential delay equations with Poisson jumps

We study a class of super-linear stochastic differential delay equations with Poisson jumps (SDDEwPJs). The convergence and rate of the convergence of the truncated Euler-Maruyama numerical solutions to SDDEwPJs are investigated under the generalized Khasminskii-type condition.     

多项Caputo分数阶随机微分方程的Euler-Maruyama方法

本文主要研究了一类多项Caputo分数阶随机微分方程的Euler-Maruyama (EM)方法,并证明了其强收敛性.具体地,我们首先构造了求解多项Caputo分数阶随机微分方程初值问题的EM方法,然后证明分数阶导数的指标满足$\frac{1}{2}<\alpha_{1}<\alpha_{2}<\cdots<\alpha_{m}<1$时,该方法是$\alpha_{m}-\alpha_{m-1}$阶强收敛的.文末的数值试验验证了理论结果的正确性.     

Strong Predictor-Corrector Approximation for Stochastic Delay Differential Equations

This paper presents a strong predictor-corrector method for the numerical solution of stochastic delay differential equations (SDDEs) of Itô-type. The method is proved to be mean-square convergent of order min{$1/2, \hat{p}$} under the Lipschitz condition and the linear growth condition, where $\hat{p}$ is the exponent of Hölder condition of the initial function. Stability criteria for this type of method are derived. It is shown that for certain choices of the flexible parameter $p$ the derived method can have a better stability property than more commonly used numerical methods. That is, for some $p$, the asymptotic MS-stability bound of the method will be much larger than that of the Euler-Maruyama method. Numerical results are reported confirming convergence properties and comparing stability properties of methods with different parameters $p$. Finally, the vectorised simulation is discussed and it is shown that this implementation is much more efficient.     

Backward Euler-Maruyama method applied to nonlinear hybrid stochastic differential equations with time-variable delay

In this paper, we consider strong convergence and almost sure exponential stability of the backward Euler-Maruyama method for nonlinear hybrid stochastic differential equations with time-variable delay. Under the local Lipschitz condition and polynomial growth condition, it is proved that the backward Euler-Maruyama method is strongly convergent. Additionally, the moment estimates and almost sure exponential stability for the analytical solution are proved. Also, under the appropriate condition, we show that the numerical solutions for the backward Euler-Maruyama methods are almost surely exponentially stable. A numerical experiment is given to illustrate the computational effectiveness and the theoretical results of the method.     

Ball Convergence Theorems for Chebyshev-Halley Method in $\mathbb{B}$-Space

A local convergence analysis of Chebyshev-Halley method having third order of convergence for approximating zero of non-linear operator $f(v)=0$ by using convex majorant function and their condition in $\mathbb{B}$-space (Banach space), is presented in this article. We give the error estimate to show the efficiency of our study. Besides, we established the relation between majorant function and Kantorovich or Smale-type result as special cases of our general theory.     

带有弱奇性核的多项分数阶非线性随机微分方程的改进Euler-Maruyama格式

针对一类带有弱奇性核的多项分数阶非线性随机微分方程构造了改进Euler-Maruyama (EM)格式,并证明了该格式的强收敛性.具体地,利用随机积分解的充分条件,将此多项分数阶随机微分方程等价地转化为随机Volterra 积分方程的形式,详细推导出对应的改进EM格式,并对该格式进行了强收敛性分析,其强收敛阶为α_m-α_m-1,其中α_i为分数阶导数的指标,且满足0<α₁<…<α_m-1<α_m<1.最后,通过数值实验验证了理论分析结果的正确性.     

On the Construction and Analysis of Finite Volume Element Schemes with Optimal $L^2$ Convergence Rate

We provide a general construction method for a finite volume element (FVE) scheme with the optimal $L^2$ convergence rate. The $k$-($k$-1)-order orthogonal condition (generalized) is proved to be a sufficient and necessary condition for a $k$-order FVE scheme to have the optimal $L^2$ convergence rate in 1D, in which the independent dual parameters constitute a ($k$-1)-dimension surface in the reasonable domain in $k$-dimension.In the analysis, the dual strategies in different primary elements are not necessarily to be the same, and they are allowed to be asymmetric in each primary element, which open up more possibilities of the FVE schemes to be applied to some complex problems, such as the convection-dominated problems. It worth mentioning that, the construction can be extended to the quadrilateral meshes in 2D. The stability and $H^1$ estimate are proved for completeness. All the above results are demonstrated by numerical experiments.     

非线性随机分数阶延迟积分微分方程Euler-Maruyama方法的强收敛性

本文研究了一类新的模型问题:非线性随机分数阶延迟积分微分方程.当方程中的漂移项和扩散项满足全局Lipschitz条件和线性增长条件时,基于压缩映射原理给出了该方程解存在唯一的充分条件.由于理论求解的困难,构造了一种数值方法(Euler-Maruyama方法),并证得强收敛阶为α-1/2,α∈(1/2,1].最后通过数值试验,验证了这一理论结果.     

Strong Convergence of the Euler-Maruyama Method for Nonlinear Stochastic Volterra Integral Equations with Time-Dependent Delay

We consider a nonlinear stochastic Volterra integral equation with time-dependent delay and the corresponding Euler-Maruyama method in this paper. Strong convergence rate (at fixed point) of the corresponding Euler-Maruyama method is obtained when coefficients $f$ and $g$ both satisfy local Lipschitz and linear growth conditions. An example is provided to interpret our conclusions. Our result generalizes and improves the conclusion in [J. Gao, H. Liang, S. Ma, Strong convergence of the semi-implicit Euler method for nonlinear stochastic Volterra integral equations with constant delay, Appl. Math. Comput., 348 (2019) 385-398.]     

Strong Convergence of the Euler-Maruyama Method for Nonlinear Stochastic Convolution Itô-Volterra Integral Equations with Constant Delay

Methodology and Computing in Applied Probability - This paper mainly focuses on the strong convergence of the Euler-Maruyama method for nonlinear stochastic convolution Itô-Volterra integral...     

Stable strong order 1.0 schemes for solving stochastic ordinary differential equations

The Balanced method was introduced as a class of quasi-implicit methods, based upon the Euler-Maruyama scheme, for solving stiff stochastic differential equations. We extend the Balanced method to introduce a class of stable strong order 1.0 numerical schemes for solving stochastic ordinary differential equations. We derive convergence results for this class of numerical schemes. We illustrate the asymptotic stability of this class of schemes is illustrated and is compared with contemporary schemes of strong order 1.0. We present some evidence on parametric selection with respect to minimising the error convergence terms. Furthermore we provide a convergence result for general Balanced style schemes of higher orders.     

On an iterative method for a class of 2 point \& 3 point nonlinear SBVPs

In this article, we propose a novel modification to Quasi-Newton method, which is now a days popularly known as variation iteration method (VIM) and use it to solve the following class of nonlinear singular differential equations which arises in chemistry $-y''(x)-\frac{\alpha}{x}y''(x)=f(x,y),~x\in(0,1),$ where $\alpha\geq1$, subject to certain two point and three point boundary conditions. We compute the relaxation parameter as a function of Bessel and the modified Bessel functions. Since rate of convergence of solutions to the iterative scheme depends on the relaxation parameter, thus we can have faster convergence. We validate our results for two point and three point boundary conditions. We allow $\partial f/\partial y$ to take both positive and negative values.     

带Markov跳时变随机种群收获系统数值解的收敛性

讨论了一类带Markov跳时变随机种群收获系统的数值解问题.利用EulerMaruyama方法给出了时变种群系统的数值解表达式,在局部Lipschitz条件下,证明了方程的数值解在均方意义下收敛于其解析解.最后,通过数值例子对所给出的结论进行了验证.     

非自治刚性随机微分方程正则EM分裂方法的收敛性和稳定性

本文研究了数值求解非自治随机微分方程的正则Euler-Maruyama分裂（CEMS）方法,该方程的漂移项系数带有刚性且允许超线性增长,扩散项系数满足全局Lipschitz条件.首先,证明了CEMS方法的强收敛性及收敛速度.其次,证明了在适当条件下CEMS方法是均方稳定的.进一步,利用离散半鞅收敛定理,研究了CEMS方法的几乎必然指数稳定性.结果表明,CEMS方法在漂移系数的刚性部分满足单边Lipschitz条件下可保持几乎必然指数稳定性.最后通过数值实验,检验了CEMS方法的有效性并证实了我们的理论结果.     

A note on order of convergence of numerical method for neutral stochastic functional differential equations

In this paper, we study the order of convergence of the Euler-Maruyama (EM) method for neutral stochastic functional differential equations (NSFDEs). Under the global Lipschitz condition, we show that the pth moment convergence of the EM numerical solutions for NSFDEs has order p/2 − 1/l for any p ? 2 and any integer l > 1. Moreover, we show the rate of the mean-square convergence of EM method under the local Lipschitz condition is 1 − ε/2 for any ε ∈  (0, 1), provided the local Lipschitz constants of the coefficients, valid on balls of radius j, are supposed not to grow faster than log j. This is significantly different from the case of stochastic differential equations where the order is 1/2.     

ON CONVERGENCE PROPERTY OF THE LANCZOS METHOD FOR SOLVING A COMPLEX SHIFTED HERMITIAN LINEAR SYSTEM

We discuss the convergence property of the Lanczos method for solving a complex shifted Hermitian linear system $(α I + H)x = f$. By showing the colinear coefficient of two system's residuals, our convergence analysis reveals that under the condition $Re(α)+ λ _{min}(H)>0$, the method converges faster than that for the real shifted Hermitian linear system $(Re(α) I+H)x=f$. Numerical experiments verify such convergence property.     

一个修正HS共轭梯度法及其收敛性

It is well-known that the direction generated by Hestenes-Stiefel (HS) conjugate gradient method may not be a descent direction for the objective function. In this paper, we take a little modification to the HS method, then the generated direction always satisfies the sufficient descent condition. An advantage of the modified Hestenes-Stiefel (MHS) method is that the scalar βkH Sffikeeps nonnegative under the weak Wolfe-Powell line search. The global convergence result of the MHS method is established under some mild conditions. Preliminary numerical results show that the MHS method is a little more efficient than PRP and HS methods.