The Milstein Scheme for Stochastic Delay Differential Equations Without Using Anticipative Calculus

The Milstein scheme is the simplest nontrivial numerical scheme for stochastic differential equations with a strong order of convergence one. The scheme has been extended to the stochastic delay differential equations but the analysis of the convergence is technically complicated due to anticipative integrals in the remainder terms. This article employs an elementary method to derive the Milstein scheme and its first order strong rate of convergence for stochastic delay differential equations.     

Strong Convergence of Euler Approximations of Stochastic Differential Equations with Delay Under Local Lipschitz Condition

The strong convergence of Euler approximations of stochastic delay differential equations is proved under general conditions. The assumptions on drift and diffusion coefficients have been relaxed to include polynomial growth and only continuity in the arguments corresponding to delays. Furthermore, the rate of convergence is obtained under one-sided and polynomial Lipschitz conditions. Finally, our findings are demonstrated with the help of numerical simulations.     

Numerical Solution to Hybrid Stochastic Differential Systems

Abstract

In this article numerical methods for solving hybrid stochastic differential systems of Itô-type are developed by piecewise application of numerical methods for SDEs. We prove a convergence result if the corresponding method for SDEs is numerically stable with uniform convergence in the mean square sense. The Euler and Runge–Kutta methods for hybrid stochastic differential equations are specifically described and the order of the error is given for the Euler method. A numerical example is given to illustrate the theory.     

Approximation of continuous time stochastic processes by a local linearization method

This paper investigates the rate of convergence of an alternative approximation method for stochastic differential equations. The rates of convergence of the one-step and multi-step approximation errors are proved to be and in the sense respectively, where is discrete time interval. The rate of convergence of the one-step approximation error is improved as compared with methods assuming the value of Brownian motion to be known only at discrete time. Through numerical experiments, the rate of convergence of the multi-step approximation error is seen to be much faster than in the conventional method.

     

线性随机变时滞微分方程指数Euler方法的收敛性和稳定性

文应用指数Euler方法研究了线性随机变时滞微分方程的收敛性和稳定性;首先,证明了指数Euler方法是$\frac{1}{2}$阶均方收敛的;其次,在解析解均方稳定的前提下,通过跟Euler-Maruyama方法比较发现指数Euler方法在大步长下依然保持解析解的均方稳定性;最后,用数值试验验证了收敛和稳定的结果.     

Strong Convergence of a Fully Discrete Scheme for Multiplicative Noise Driving SPDEs with Non-Globally Lipschitz Continuous Coefficients

This work investigates strong convergence of numerical schemes for nonlinear multiplicative noise driving stochastic partial differential equations under some weaker conditions imposed on the coefficients avoiding the commonly used global Lipschitz assumption in the literature. Space-time fully discrete scheme is proposed, which is performed by the finite element method in space and the implicit Euler method in time. Based on some technical lemmas including regularity properties for the exact solution of the considered problem, strong convergence analysis with sharp convergence rates for the proposed fully discrete scheme is rigorously established.     

Extending convergence theory for nonlinear stiff problems part I

Existing convergence concepts for the analysis of discretizations of nonlinear stiff problems suffer from considerable drawbacks. Our intention is to extend the convergence theory to a relevant class of nonlinear problems, where stiffness is axiomatically characterized in natural geometric terms.Our results will be presented in a series of papers. In the present paper (Part I) we motivate the need for such an extension of the existing theory, and our approach is illustrated by means of a convergence argument for the Implicit Euler scheme.     

Two-scale Convergence and Homogenization for a Class of Quasilinear Elliptic Equations

By use of Fourier analysis techniques, we obtain some new properties of the almost-periodic functions and extend the two-scale convergence method in the homogenization theory to the case of almost-periodic oscillations. Then, we use some new techniques to study the homogenization for quasilinear elliptic equations with almostperiodic coefficients: div a(x,x/ε, u, Du) = f(x) in Ω and obtain the weak convergence and corrector result.     

Homogenization of Optimal Control Problems for Functional Differential Equations

The paper deals with the variational convergence of a sequence of optimal control problems for functional differential state equations with deviating argument. Variational limit problems are found under various conditions of convergence of the input data. It is shown that, upon sufficiently weak assumptions on convergence of the argument deviations, the limit problem can assume a form different from that of the whole sequence. In particular, it can be either an optimal control problem for an integro-differential equation or a purely variational problem. Conditions are found under which the limit problem preserves the form of the original sequence.     

Penalty method for solving a class of stochastic differential variational inequalities with an application

The aim of this paper is to study the penalty method for solving a class of stochastic differential variational inequalities (SDVIs). The penalty problem for solving SDVIs is first constructed and the convergence of the sequences generated by the penalty problem is proved under some mild conditions. As an application, the convergence of the sequences generated by the penalty problem is obtained for solving a stochastic migration equilibrium problem with movement cost.     

Convergence of Weighted Empirical Measures

This article considers a system with infinitely many interacting particles, starting with a system of n interacting particles that is described by a system of n stochastic differential equations for the time-varying locations and weights. Any particle in the system interacts with others through the weighted empirical measure Uⁿ formed by the sum of weighted Dirac measures on the n particles. Weak convergence of the weighted empirical measure is studied under suitable conditions, such as bounded initial values, and linear growth of drift and diffusion coefficients. Thereafter, the limit of the weighted empirical measures is identified to be a martingale solution of the infinite interacting system.     

Convergence and stability of the semi-implicit Euler method for a linear stochastic differential delay equation

The paper deals with convergence and stability of the semi-implicit Euler method for a linear stochastic differential delay equation. It is proved that the semi-implicit Euler method is convergent with strong order . The conditions under which the method is MS-stable and GMS-stable are determined and the numerical experiments are given.     

一类求解奇异延迟微分方程的两步连续Runge-Kutta方法的收敛性

本文给出了一类求解延迟落在当前积分步内延迟微分方程的两步连续Runge-Kutta方法。在一定条件下我们证明了方法收敛性,数值试验表明方法是有效的。     

Existence and uniqueness of the solutions and convergence of semi-implicit Euler methods for stochastic pantograph equations

In this paper the sufficient conditions of existence and uniqueness of the solutions for stochastic pantograph equation are given, i.e., the local Lipschitz condition and the linear growth condition. Under the Lipschitz condition and the linear growth condition it is proved that the semi-implicit Euler method is convergence with strong order .     

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.     

Stochastic partitioned averaged vector field methods for stochastic differential equations with a conserved quantity

In this paper, stochastic differential equations in the Stratonovich sense with a conserved quantity are considered. A stochastic partitioned averaged vector field method is proposed and analyzed. We prove this numerical method is able to preserve the conserved quantity of the original system. Then the convergence analysis is carried out in detail and we derive the method is convergent with order $1$ in the mean-square sense. Finally some numerical examples are reported to verify the effectiveness and flexibility of the proposed method.     

CONVERGENCE ANALYSIS OF PARAREAL ALGORITHM BASED ON MILSTEIN SCHEME FOR STOCHASTIC DIFFERENTIAL EQUATIONS

In this paper, we propose a parareal algorithm for stochastic differential equations (SDEs), which proceeds as a two-level temporal parallelizable integrator with the Milstein scheme as the coarse propagator and the exact solution as the fine propagator. The convergence order of the proposed algorithm is analyzed under some regular assumptions. Finally, numerical experiments are dedicated to illustrating the convergence and the convergence order with respect to the iteration number $k$, which show the efficiency of the proposed method.     

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.     

Strong convergence of a stochastic Rosenbrock-type scheme for the finite element discretization of semilinear SPDEs driven by multiplicative and additive noise

This paper aims to investigate the numerical approximation of a general second order parabolic stochastic partial differential equation(SPDE) driven by multiplicative and additive noise. Our main interest is on such SPDEs where the nonlinear part is stronger than the linear part, usually called stochastic dominated transport equations. Most standard numerical schemes lose their good stability properties on such equations, including the current linear implicit Euler method. We discretize the SPDE in space by the finite element method and propose a novel scheme called stochastic Rosenbrock-type scheme for temporal discretization. Our scheme is based on the local linearization of the semi-discrete problem obtained after space discretization and is more appropriate for such equations. We provide a strong convergence of the new fully discrete scheme toward the exact solution for multiplicative and additive noise and obtain optimal rates of convergence. Numerical experiments to sustain our theoretical results are provided.     

On the approximation of stochastic differential equations

The stability properties of stochastic differential equations with respetct to the perturbation of the coefficients and of the driving processes are investigated in the topology of uniform convergence in probability