改进欧拉格式求解随机微分方程的收敛性

采用改进的欧拉格式求解随机微分方程,当方程的偏移系数和扩散系数均满足全局Lipschitz条件和线性增长条件时,证明改进格式的强收敛的阶是1/2.     

改进欧拉格式求解随机微分方程的收敛性

采用改进的欧拉格式求解随机微分方程,当方程的偏移系数和扩散系数均满足全局Lipschitz条件和线性增长条件时,证明改进格式的强收敛的阶是1/2.     

有限体积法定价美式期权

讨论美式期权定价的有限体积法.采用投影超松弛迭代法求解隐式欧拉和CrankNicolson有限体积格式离散Black-Scholes偏微分方程得到的线性互补问题.数值实验结果表明,两种有限体积格式都是有效的,而Crank-Nicolson格式的数值效果要优于隐式欧拉格式.     

线性随机时滞微分方程裂步Milstein方法的收敛性(英文)

给出一个新的求解线性随机时滞微分方程的显式分裂步长Milstein格式.运用ItoTaylor展开式证明该格式相对于已有的求解随机时滞微分方程的分裂步长方法而言具有更好的收敛性.数值实验验证了理论分析的正确性.     

带有弱奇性核的多项分数阶非线性随机微分方程解的适定性

本文主要研究一类带有多项分数阶Caputo导数的非线性随机微分方程初值问题的解的适定性.具体地,首先把多项分数阶随机微分方程等价地转化为随机Volterra积分方程;然后,给出了该随机积分方程的Euler-Maruyama (EM)格式;最后,借助于该EM格式,证明了多项分数阶随机微分方程的解的适定性.     

四阶R-K方法中一类新算法的分析

对常微分方程初值问题数值计算中的四阶R-K方法首次具体给出了一般格式中的参数所满足的方程,并提出了新的计算格式,这些新算法对某些初值问题其整体截断误差有明显的减少.这对常微分方程初值问题在社会、经济、生态等领域中的广泛应用将提供有益的新算法.     

中立型随机比例微分方程的数值解的指数稳定性（英文）

本文主要利用半鞅收敛定理,研究中立型随机比例微分方程的数值稳定性.该文建立了线性的和非线性的中立型随机比例微分方程新的细则,我们将证明,在线性增长条件下,欧拉方法可以保留中立型随机比例微分方程的几乎处处指数稳定性,并且反向的欧拉方法能保留非线性的中立型随机比例微分方程的几乎处处指数稳定性.     

线性算子方程数值解的高精度算法

本文探讨一般线性算子方程数值解的高精度算法.给出了迭代校正计算格式及误差估计.用于微分方程及积分方程,并作了精度和效率分析.     

二维热传导方程的三层显式差分格式

对二维热传导方程构造了一个稳定的三层显式差分格式求其数值解,其背景源于高维热力学反问题迭代算法中对正问题小计算量算法的需求。首先建立一个含参数的一般差分格式去逼近微分方程,并得到了最优截断误差。然后导出了参数应满足的条件以保证差分格式的稳定性。最后给出了数值的例子并和其它算法进行比较,说明了格式在精度上的有效性和计算量上的优越性。     

具有马尔可夫调制的随机微分方程数值解的收敛性

本文在无穷维Hilbert空间中研究了一类具有马尔可夫调制的随机微分方程(SDEwMSs).在一般情况下SDEwMSs没有解析解.因此合适的数值逼近法,例如欧拉法,就是在研究它们性质时所采用的重要工具.本文在较弱的条件下不仅证明了欧拉近似解收敛于SDEwMSs的精确解(分析解),而且给出了欧拉近似阶的界.     

Approximation for non-smooth functionals of stochastic differential equations with irregular drift

This paper aims at developing a systematic study for the weak rate of convergence of the Euler–Maruyama scheme for stochastic differential equations with very irregular drift and constant diffusion coefficients. We apply our method to obtain the rates of approximation for the expectation of various non-smooth functionals of both stochastic differential equations and killed diffusion. We also apply our method to the study of the weak approximation of reflected stochastic differential equations whose drift is Hölder continuous.     

Banach空间中带Poisson跳的随机种群方程的数值分析

本文研究Banach空间中带Poisson跳的随机种群方程,通过离散使之成为随机微分方程,进而运用显式Euler公式来分析其数值解与解析解的误差.     

Rate of convergence in the Euler scheme for stochastic differential equations with non-Lipschitz diffusion and Poisson measure

We study the rate of convergence and some other properties of the Euler scheme for stochastic differential equations with non-Lipschitz diffusion and Poisson measure.     

A novel approach to construct numerical methods for stochastic differential equations

In this paper we propose a new numerical method for solving stochastic differential equations (SDEs). As an application of this method we propose an explicit numerical scheme for a super linear SDE for which the usual Euler scheme diverges.     

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.     

倒向随机方程期权定价模型的一类随机算法

期权定价问题可以转化为对倒向随机微分方程的求解,进而转化为对相应抛物型偏微分方程的求解.为了求解与倒向随机微分方程相应的二阶拟线性抛物型微分方程初值问题,引入一类新的随机算法-分层方法取代传统的确定性数值算法.这种数值方法理论上是通过弱显式欧拉法,离散其相应随机系统解的概率表示而得到.该随机算法的收敛性在文中得到证明,其稳定性是自然的.并构造了易于数值实现的基于插值的算法,实证研究说明这种算法能很好地提供期权定价模型的数值模拟.     

The Euler Scheme for Feller Processes

We consider the Euler scheme for stochastic differential equations with jumps, whose intensity might be infinite and the jump structure may depend on the position. This general type of SDE is explicitly given for Feller processes and a general convergence condition is presented.

In particular, the characteristic functions of the increments of the Euler scheme are calculated in terms of the symbol of the Feller process in a closed form. These increments are increments of Lévy processes and, thus, the Euler scheme can be used for simulation by applying standard techniques from Lévy processes.     

The Pathwise Numerical Approximation of Stationary Solutions of Semilinear Stochastic Evolution Equations

Under a one-sided dissipative Lipschitz condition on its drift, a stochastic evolution equation with additive noise of the reaction-diffusion type is shown to have a unique stochastic stationary solution which pathwise attracts all other solutions. A similar situation holds for each Galerkin approximation and each implicit Euler scheme applied to these Galerkin approximations. Moreover, the stationary solution of the Euler scheme converges pathwise to that of the Galerkin system as the stepsize tends to zero and the stationary solutions of the Galerkin systems converge pathwise to that of the evolution equation as the dimension increases. The analysis is carried out on random partial and ordinary differential equations obtained from their stochastic counterparts by subtraction of appropriate Ornstein-Uhlenbeck stationary solutions.     

Weak Euler Approximation for Itô Diffusion and Jump Processes

This article studies the rate of convergence of the weak Euler approximation for Itô diffusion and jump processes with Hölder-continuous generators. It covers a number of stochastic processes including the nondegenerate diffusion processes and a class of stochastic differential equations driven by stable processes. To estimate the rate of convergence, the existence of a unique solution to the corresponding backward Kolmogorov equation in Hölder space is first proved. It then shows that the Euler scheme yields positive weak order of convergence.     

A Runge–Kutta type scheme for nonlinear stochastic partial differential equations with multiplicative trace class noise

In this paper a new Runge–Kutta type scheme is introduced for nonlinear stochastic partial differential equations (SPDEs) with multiplicative trace class noise. The proposed scheme converges with respect to the computational effort with a higher order than the well-known linear implicit Euler scheme. In comparison to the infinite dimensional analog of Milstein type scheme recently proposed in Jentzen and Röckner (2012), our scheme is easier to implement and needs less computational effort due to avoiding the derivative of the diffusion function. The new scheme can be regarded as an infinite dimensional analog of Runge–Kutta method for finite dimensional stochastic ordinary differential equations (SODEs). Numerical examples are reported to support the theoretical results.