Stochastic Differential Algebraic Equations of index 1 and Applications in Circuit Simulation

Preprint series: Institut für Mathematik, Humboldt-Universität zu Berlin (ISSN 0863-0976)

MSC 2000

65C30 Stochastic differential and integral equations

94C99 None of the above, but in this section

CR: G3

Abstract
In this paper we deal with differential-algebraic 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 differential-algebraic 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 DAE-index 1. In a second part we analyze discretization methods. Due to the differential-algebraic 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 drift-implicit discretization schemes for SDEs is proved. Then, we apply the drift-implicit Euler scheme, the split-step backward Euler scheme, the trapezoidal rule and the drift-implicit 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 network-topology.

Keywords:numerical methods, stochastic differential-algebraic equations, transient noise simulation