Evelyn Buckwar [Homepage] and Thorsten Sickenberger [Homepage]
Maxwell Institute for Mathematical Sciences and
Heriot-Watt University Edinburgh, School of Mathematical and Computer Sciences, Department of Mathematics, Edinburgh.

Software (MATLAB® files) for
Asymptotic mean-square stability analysis of linear systems of stochastic differential equations (SDEs)
and their numerical approximation

based on the paper:
E. Buckwar, T. Sickenberger:
A structural analysis of asymptotic mean-sqaure stability for multi-dimensional linear stochastic differential systems,
Heriot-Watt Mathematics Report HWM10-21, 2010.
    Abstract We are concerned with a linear mean-square stability analysis of numerical methods applied to systems of stochastic differential equations (SDEs) and, in particular, consider the Theta-Maruyama and the Theta-Milstein method in this context. We propose a technique, based on the vectorisation of matrices and the Kronecker product, to deal with the matrix expressions arising in this analysis and provide the explicit structure of the stability matrices in the general case of linear systems of SDEs. For a set of simple test SDE systems, incorporating different noise structures but only a few parameters, we apply the general results and provide visual and numerical comparisons of the stability properties of the two methods.
Keywords Asymptotic mean-square stability • Theta-Maruyama method • Theta-Milstein method • Systems of stochastic differential equations • LLinear stability analysis
Software download You can test the asymptotic mean-square stability of your own d-dimensional linear system of SDEs having m multiplicative noise sources. Download the following files and insert your drift and diffusion matrices (see the "else" part of the first loop in the main file). Alternatively you can run the test SDE systems of our paper.
main_stability.m - main program file (MATLAB®),
  defines test-equations and writes outputs
stabanalysis.m - function file (MATLAB®),
  analysis the system and computes the asymptotic mean-square stability of the zero solution
stabcalculation.m - function file (MATLAB®),
  computes the asymptotic mean-square stability of Theta-Maruyama and Theta-Milstein approximations
  for different values of the step-size and the method parameter Theta.
Remark You might be also interested in
A comparative linear mean-square stability analysis of Maruyama- and Milstein-type methods (by Evelyn Buckwar and Thorsten Sickenberger),
Heriot-Watt Mathematics Report HWM09-13, 2009.
Contact • Evelyn Buckwar [Homepage]
  e-mail: e.buckwar@hw.ac.uk

• Thorsten Sickenberger [Homepage]
  e-mail: t.sickenberger@hw.ac.uk
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