Convergence of numerical schemes for the solution of parabolic stochastic partial differential equations

We consider the numerical solution of the stochastic partial differential equation , where is space-time white noise, using finite differences. For this equation Gyöngy has obtained an estimate of the rate of convergence for a simple scheme, based on integrals of over a rectangular grid. We investigate the extent to which this order of convergence can be improved, and find that better approximations are possible for the case of additive noise ( ) if we wish to estimate space averages of the solution rather than pointwise estimates, or if we are permitted to generate other functionals of the noise. But for multiplicative noise ( ) we show that no such improvements are possible.

     

Weak exponential schemes for stochastic differential equations with additive noise

^** Email: cmora{at}ing-mat.udec.cl This paper develops weak exponential schemes for the numericalsolution of stochastic differential equations (SDEs) with additivenoise. In particular, this work provides first and second-ordermethods which use at each iteration the product of the exponentialof the Jacobian of the drift term with a vector. The articlealso addresses the rate of convergence of the new schemes. Moreover,numerical experiments illustrate that the numerical methodsintroduced here are a good alternative to the standard integratorsfor the long time integration of SDEs whose solutions by thecommon explicit schemes exhibit instabilities.     

Error and convergence of two numerical schemes for stochastic differential equations

We examine the convergence and error rate of two stochastic numerical schemes using the method of proof used by G. N. Mil'shtein 1 . © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006.     

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.     

Expansion of the global error for numerical schemes solving stochastic differential equations

Given the solution (X_t ) of a Stochastic Differential System, two situat,ions are considered: computat,ion of Ef(X_t ) by a Monte–Carlo method and, in the ergodic case, integration of a function f w.r.t. the invariant probability law of (X_t ) by simulating a simple t,rajectory.

For each case it is proved the expansion of the global approximat,ion error—for a class of discret,isat,ion schemes and of funct,ions f—in powers of the discretisation step size, extending in the fist case a result of Gragg for deterministic O.D.E.

Some nn~nerical examples are shown to illust,rate the applicat,ion of extrapolation methods, justified by the foregoing expansion, in order to improve the approximation accuracy     

Convergence rate of strong Local Linearization schemes for stochastic differential equations with additive noise

There is a variety of strong Local Linearization (LL) schemes for the numerical integration of stochastic differential equations with additive noise, which differ with respect to the algorithm that is used in the numerical implementation of the strong Local Linear discretization. However, in contrast with the Local Linear discretization, the convergence rate of the LL schemes has not been studied so far. In this paper, two general theorems about this matter are presented and, with their support, additional results are derived for some particular schemes. As a direct application, the convergence rate of some strong LL schemes for SDEs with jumps is briefly expounded as well.     

Weak exponential stability of stochastic differential equations

In this paper we introduce weak exponential stability of stochastic differential equations. In particular, we introduce weak exponential stability in mean, weak exponential asymptotical stability in mean and weak uniform asymptotical stability in mean. We also derive some results related to the above concepts     

Weak solutions to gamma-driven stochastic differential equations

We study a stochastic differential equation driven by a gamma process, for which we give results on the existence of weak solutions under conditions on the volatility function. To that end we provide results on the density process between the laws of solutions with different volatility functions.     

Two-step strong order 1.5 schemes for stochastic differential equations

A general class of linear two-step schemes for solving stochastic differential equations is presented. Necessary and sufficient conditions on its parameters to obtain mean square order 1.5 are derived. Then the linear stability of the schemes is investigated. In particular, among others, the stability regions of generalizations of the classical two-step schemes Adams-Bashford, Adams-Moulton, and BDF are obtained.     

Convergence of numerical solutions to neutral stochastic delay differential equations with Markovian switching

Recently, numerical solutions of stochastic differential equations have received a great deal of attention. It is surprising that there are not any numerical methods established for neutral stochastic delay differential equations yet. In the paper, the Euler–Maruyama method for neutral stochastic delay differential equations is developed. The key aim is to show that the numerical solutions will converge to the true solutions under the local Lipschitz condition.     

Weak solution for stochastic differential equations with terminal conditions

The notion of weak solution for stochastic differential equation with terminal conditions is introduced. By Girsanov transformation, the equivalence of existence of weak solutions for two-type equations is established. Several sufficient conditions for the existence of the weak solutions for stochastic differential equation with terminal conditions are obtained, and the solution existence condition for this type of equations is relaxed. Finally, an example is given to show that the result is an essential extension of the one under Lipschitz condition ong with respect to (Y,Z).     

Symmetry preserving numerical schemes for partial differential equations and their numerical tests

The method of equivariant moving frames is used to construct symmetry preserving finite difference schemes of partial differential equations invariant under finite-dimensional symmetry groups. Invariant numerical schemes for a heat equation with logarithmic source and the spherical Burgers' equation are obtained. Numerical tests show how invariant schemes can be more accurate than standard discretizations.     

Weak order stochastic Runge–Kutta methods for commutative stochastic differential equations

A new explicit stochastic Runge–Kutta scheme of weak order 2 is proposed under a commutativity condition, which is derivative-free and which attains order 4 for ordinary differential equations. The weak order conditions are derived by utilizing multi-colored rooted tree analysis and a solution is found in a transparent way. The scheme is compared with other derivative-free and weak second order schemes in numerical experiments.     

Mean-square stability analysis of numerical schemes for stochastic differential systems

For ordinary differential systems, the study of A-stability for a numerical method reduces to the scalar case by means of a transformation that uncouples the linear test system as well as the difference system provided by the method. For stochastic differential equations (SDEs), mean-square stability (MS-stability) has been successfully proposed as the generalization of A-stability, and numerical MS-stability has been analyzed for one-dimensional equations. However, unlike the deterministic case, the extension of this analysis to multi-dimensional systems is not straightforward. In this paper we give necessary and sufficient conditions for the MS-stability of multi-dimensional systems with one Wiener noise. The criterion presented does not depend on any norm. Based on the Routh–Hurwitz theorem, we offer a particular criterion of MS-stability for two-dimensional systems in terms of their coefficients. In addition, a counterpart criterion of MS-stability is given for numerical schemes applied to multi-dimensional systems. The MS-stability behavior of a stochastic numerical method is determined by the comparison of its stability region with the stability region of the system. As an application, the numerical MS-stability of θ$\theta$-methods applied to bi-dimensional systems is investigated.     

Almost sure exponential stability of backward Euler-Maruyama discretizations for hybrid stochastic differential equations

This is a continuation of the first author’s earlier paper [1] jointly with Pang and Deng, in which the authors established some sufficient conditions under which the Euler-Maruyama (EM) method can reproduce the almost sure exponential stability of the test hybrid SDEs. The key condition imposed in [1] is the global Lipschitz condition. However, we will show in this paper that without this global Lipschitz condition the EM method may not preserve the almost sure exponential stability. We will then show that the backward EM method can capture the almost sure exponential stability for a certain class of highly nonlinear hybrid SDEs.     

Second-order schemes for solving decoupled forward backward stochastic differential equations

In this paper,by using trapezoidal rule and the integration-by-parts formula of Malliavin calculus,we propose three new numerical schemes for solving decoupled forward-backward stochastic differential equations.We theoretically prove that the schemes have second-order convergence rate.To demonstrate the effectiveness and the second-order convergence rate,numerical tests are given.     

Higher-order semi-implicit Taylor schemes for Itô stochastic differential equations

The paper considers the derivation of families of semi-implicit schemes of weak order N=3.0 (general case) and N=4.0 (additive noise case) for the numerical solution of Itô stochastic differential equations. The degree of implicitness of the schemes depends on the selection of N parameters which vary between 0 and 1 and the families contain as particular cases the 3.0 and 4.0 weak order explicit Taylor schemes. Since the implementation of the multiple integrals that appear in these theoretical schemes is difficult, for the applications they are replaced by simpler random variables, obtaining simplified schemes. In this way, for the multidimensional case with one-dimensional noise, we present an infinite family of semi-implicit simplified schemes of weak order 3.0 and for the multidimensional case with additive one-dimensional noise, we give an infinite family of semi-implicit simplified schemes of weak order 4.0. The mean-square stability of the 3.0 family is analyzed, concluding that, as in the deterministic case, the stability behavior improves when the degree of implicitness grows. Numerical experiments confirming the theoretical results are shown.