带跳的随机微分方程的Euler折线逼近

熟知当随机微分方程的系数不满足Lipschitz条件,而仅满足单调性条件时,我们无法用Picard迭代法证明其解的存在性. Krylov为此对Brown运动驱动的此类方程用Euler折线逼近法证明了解的存在性.本文将Krylov的结果推广到带跳的随机微分方程,证明了Euler折线逼近的收敛性.这一结果是研究带跳的随机发展方程的基础,且对随机微分方程的数值计算有用.     

延迟微分方程隐式Euler方法的收敛性

本文讨论了用隐式Euler方法求解一类延迟量满足Lipschitz条件且Lipschitz常数小于1的非线性变延迟微分方程初值问题的收敛性.获得了带线性插值的隐式Euler方法的收敛性结果.     

带Poisson跳和Markovian调制的中立型随机延迟微分方程的数值解的收敛性

本文研究带Poisson跳和Markovian调制的中立型随机微分方程的数值解的收敛性质.用数值逼近方法求此微分方程的解,并证明了Euler近似解在此线性增长条件和全局Lipschitz条件更弱的条件下仍均方收敛于此方程的解析解.     

非光滑方程组牛顿法的全局收敛性分析

通过定义一种新的*-微分,本文给出了局部Lipschitz非光滑方程组的牛顿法,并对其全局收敛性进行了研究.该牛顿法结合了非光滑方程组的局部收敛性和全局收敛性.最后,我们把这种牛顿法应用到非光滑函数的光滑复合方程组问题上,得到了较好的收敛性.     

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.     

非Lipschitz条件下的包含下微分算子的带跳倒向随机微分方程(英文)

本文在非Lipschitz系数下,考虑了一类多值的倒向随机微分方程.利用极大单调算子的Yosida估计和倒向随机微分方程在非Lipschitz条件下解的存在唯一性,获得了多值带跳的倒向随机微分方存在唯一解的结论.     

带Markov状态转换的跳扩散方程的数值解

研究了一类带Markov状态转换的跳扩散方程的数值解的问题,为讨论这类方程精确解的数值计算问题,我们给出了一种基于Euler格式的方程解的跳适应算法,并在一定的条件下,证明了基于这种新的跳适应算法所得到的方程的数值解是收敛于它的精确解,同时还给出了数值解收敛到其精确解的收敛阶数.最后,本文通过两个例子说明了这种跳适应算法的计算有效性.     

一类随机延迟微分方程截断型theta-EM算法的收敛性

本文主要考察随机延迟微分方程截断型theta-EM (Euler-Maruyama)算法的强收敛性问题.将截断型EM算法推广到一般形式,提出截断型theta-EM算法,并讨论随机延迟微分方程在非全局Lipschitz条件下的强收敛率,得到其强收敛阶数.     

P0函数非线性互补问题的一步非内点连续方法的收敛性

本文对于P0函数非线性互补问题提出了一个基于Kanzow光滑函数的一步非内点连续方法,在适当的假设条件下,证明了方法的全局线性及局部二次收敛性.特别,在方法的全局线性收敛性的分析中,不需要假定非线性互补问题的函数的Jacobi阵是Lipschitz连续的.文献中为了得到非内点连续方法的全局线性收敛性,这一假定是被广泛使用的.本文提出的方法在每一次迭代只须解一个线性方程式组.     

非线性互补问题光滑化拟牛顿算法的收敛性分析

给出求解p_0函数非线性互补问题光滑化拟牛顿算法,在p_0函数非线性互补问题有非空有界解集且F'是Lipschitz连续的条件下,证明了算法的全局收敛性.全局收敛性的主要特征是不需要提前假设水平集是有界的.     

A transformed jump-adapted backward Euler method for jump-extended CIR and CEV models

A novel time-stepping scheme, called transformed jump-adapted backward Euler method, is developed in this paper to simulate a class of jump-extended CIR and CEV models. The proposed scheme is able to preserve the positivity of the underlying problems. Furthermore, its strong convergence rate of order one is recovered for the considered models with non-Lipschitz diffusion coefficients. Numerical examples are finally reported to confirm our theoretical findings.     

Strong convergence rates for backward Euler on a class of nonlinear jump-diffusion problems

We generalise the current theory of optimal strong convergence rates for implicit Euler-based methods by allowing for Poisson-driven jumps in a stochastic differential equation (SDE). More precisely, we show that under one-sided Lipschitz and polynomial growth conditions on the drift coefficient and global Lipschitz conditions on the diffusion and jump coefficients, three variants of backward Euler converge with strong order of one half. The analysis exploits a relation between the backward and explicit Euler methods.     

Strong convergence of split-step backward Euler method for stochastic differential equations with non-smooth drift

In this paper, we are concerned with the numerical approximation of stochastic differential equations with discontinuous/nondifferentiable drifts. We show that under one-sided Lipschitz and general growth conditions on the drift and global Lipschitz condition on the diffusion, a variant of the implicit Euler method known as the split-step backward Euler (SSBE) method converges with strong order of one half to the true solution. Our analysis relies on the framework developed in [D. J. Higham, X. Mao and A. M. Stuart, Strong convergence of Euler-type methods for nonlinear stochastic differential equations, SIAM Journal on Numerical Analysis, 40 (2002) 1041-1063] and exploits the relationship which exists between explicit and implicit Euler methods to establish the convergence rate results.     

Numerical Solutions of Stochastic Differential Delay Equations with Jumps

Abstract

In this article, we investigate the strong convergence of the Euler–Maruyama method and stochastic theta method for stochastic differential delay equations with jumps. Under a global Lipschitz condition, we not only prove the strong convergence, but also obtain the rate of convergence. We show strong convergence under a local Lipschitz condition and a linear growth condition. Moreover, it is the first time that we obtain the rate of the strong convergence under a local Lipschitz condition and a linear growth condition, i.e., if the local Lipschitz constants for balls of radius R are supposed to grow not faster than log R.     

Convergence of the Tamed Euler Method for Stochastic Differential Equations with Piecewise Continuous Arguments Under Non-global Lipschitz Continuous Coefficients

In this paper, we deal with the strong convergence of numerical methods for stochastic differential equations with piecewise continuous arguments (SEPCAs) with at most polynomially growing drift coefficients and global Lipschitz continuous diffusion coefficients. An explicit and time-saving tamed Euler method is used to solve this type of SEPCAs. We show that the tamed Euler method is bounded in pth moment. And then the convergence of the tamed Euler method is proved. Moreover, the convergence order is one-half. Several numerical simulations are shown to verify the convergence of this method.     

Almost sure exponential stability of numerical solutions for stochastic delay differential equations with jumps

This paper deals with the almost sure exponential stability of the Euler-type methods for nonlinear stochastic delay differential equations with jumps by using the discrete semimartingale convergence theorem. It is shown that the explicit Euler method reproduces the almost sure exponential stability under an additional linear growth condition. By replacing the linear growth condition with the one-sided Lipschitz condition, the backward Euler method is able to reproduce the stability property.     

The tamed Milstein method for commutative stochastic differential equations with non-globally Lipschitz continuous coefficients

For stochastic differential equations (SDEs) with a superlinearly growing and globally one-sided Lipschitz continuous drift coefficient, the classical explicit Euler scheme fails to converge strongly to the exact solution. Recently, an explicit strongly convergent numerical scheme, called the tamed Euler method, has been proposed in [8] for such SDEs. Motivated by their work, we here introduce a tamed version of the Milstein scheme for SDEs with commutative noise. The proposed method is also explicit and easily implementable, but achieves higher strong convergence order than the tamed Euler method does. In recovering the strong convergence order one of the new method, new difficulties arise and kind of a bootstrap argument is developed to overcome them. Finally, an illustrative example confirms the computational efficiency of the tamed Milstein method compared to the tamed Euler method.     

On the rate of convergence of simple and jump-adapted weak Euler schemes for Lévy driven SDEs

The paper studies the rate of convergence of a weak Euler approximation for solutions to possibly completely degenerate SDEs driven by Lévy processes, with Hölder-continuous coefficients. It investigates the dependence of the rate on the regularity of coefficients and driving processes and its robustness to the approximation of the increments of the driving process. A convergence rate is derived for some approximate jump-adapted Euler scheme as well.     

Compensated split-step balanced methods for nonlinear stiff SDEs with jump-diffusion and piecewise continuous arguments

This paper deals with numerical solutions of nonlinear stiff stochastic differential equations with jump-diffusion and piecewise continuous arguments. By combining compensated split-step methods and balanced methods, a class of compensated split-step balanced (CSSB) methods are suggested for solving the equations. Based on the one-sided Lipschitz condition and local Lipschitz condition, a strong convergence criterion of CSSB methods is derived. It is proved under some suitable conditions that the numerical solutions produced by CSSB methods can preserve the mean-square exponential stability of the corresponding analytical solutions. Several numerical examples are presented to illustrate the obtained theoretical results and the effectiveness of CSSB methods. Moreover, in order to show the computational advantage of CSSB methods, we also give a numerical comparison with the adapted split-step backward Euler methods with or without compensation and tamed explicit methods.

     

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.