A comparative linear mean-square stability analysis of Maruyama- and Milstein-type methods. (arXiv:0912.1968v1 [math.NA])

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. As a simple extension of the standard geometric
Brownian motion as a test equation for the linear stability analysis, we
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.

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Original source : http://arxiv.org/abs/0912.1968...