Renate Winkler
Preprint series: Institut für Mathematik, HumboldtUniversität
zu Berlin (ISSN 08630976)
MSC 2000

65C30 Stochastic differential and integral equations

94C99 None of the above, but in this section
ZDM:
N40
CR: G3
Abstract
In
this paper we deal with differentialalgebraic equations driven by Gaussian
white noise. In a first part we use the theory of stochastic differential
equations (SDEs) as well as the theory of differentialalgebraic equations
(DAEs) for a mathematically rigorous formulation of such problems and give
the necessary analytical theory for the existence and uniqueness of strong
solutions for systems of DAEindex 1. In a second part we analyze discretization
methods. Due to the differentialalgebraic structure implicit methods become
necessary and computational errors have to be taken into account more carefully.
For that purpose a result concerning the mean square numerical stability
for general driftimplicit discretization schemes for SDEs is proved. Then,
we apply the driftimplicit Euler scheme, the splitstep backward Euler
scheme, the trapezoidal rule and the driftimplicit Milstein scheme directly
to the stochastic DAE, estimate the influence of errors and and prove that
the convergence properties of these methods known for SDEs are preserved.
We show how the theory applies to the transient noise simulation of electronic
circuits and express the necessary conditions in terms of the networktopology.
Keywords:numerical
methods, stochastic differentialalgebraic equations, transient noise simulation