A comparative linear mean-square stability analysis of Maruyama- and Milstein-type methods

In this article we compare the mean-square stability properties of the Theta-Maruyama and Theta-Milstein method that are used to solve stochastic differential equations. For the linear stability analysis, we propose an extension of the standard geometric Brownian motion as a test equation and consider a scalar linear test equation with several multiplicative noise terms. This test equation allows to begin investigating the influence of multi-dimensional noise on the stability behaviour of the methods while the analysis is still tractable. Our findings include: (i) the stability condition for the Theta-Milstein method and thus, for some choices of Theta, the conditions on the step-size, are much more restrictive than those for the Theta-Maruyama method; (ii) the precise stability region of the Theta-Milstein method explicitly depends on the noise terms. Further, we investigate the effect of introducing partially implicitness in the diffusion approximation terms of Milstein-type methods, thus obtaining the possibility to control the stability properties of these methods with a further method parameter Sigma. Numerical examples illustrate the results and provide a comparison of the stability behaviour of the different methods.Comment: 19 pages, 10 figure

Mean-square convergence of stochastic multi-step methods with variable step-size

We study mean-square consistency, stability in the mean-square sense and mean-square convergence of drift-implicit linear multi-step methods with variable step-size for the approximation of the solution of Ito stochastic differential equations. We obtain conditions that depend on the step-size ratios and that ensure mean-square convergence for the special case of adaptive two-step Maruyama schemes. Further, in the case of small noise we develop a local error analysis with respect to the h-ε approach and we construct some stochastic linear multi-step methods with variable step-size that have order 2 behavior if the noise is small enough

Local error estimates for moderately smooth problems

The paper consists of two parts. In the first part of the paper, we proposed a procedure to estimate local errors of low order methods applied to solve initial value problems in ordinary differential equations (ODEs) and index 1 differential-algebraic equations (DAEs). Based on the idea of Defect Correction we developed local error estimates for the case when the problem data is only moderately smooth, which is typically the case in stochastic differential equations. In this second part, we will consider the estimation of local errors in context of mean-square convergent methods for stochastic differential equations (SDEs) with small noise and index 1 stochastic differential-algebraic equations (SDAEs). Numerical experiments illustrate the performance of the mesh adaptation based on the local error estimation developed in this paper

Local Error Estimates for Moderately Smooth ODEs and DAEs

We discuss an error estimation procedure for the local errors of low order methods applied to solve initial value problems in ordinary differential equations (ODEs) and index 1 differential-algebraic equations (DAEs). The proposed error estimation strategy is based on the principle of Defect Correction. Here, we present how this idea can be adapted for the estimation of local errors in case when the problem data is only moderately smooth. Moreover, we illustrate the performance of the mesh adaptation based on the error estimation developed in this paper