Stochastic differential algebraic equations of index 1 and applications in circuit simulation

Abstract

We discuss differential-algebraic equations driven by Gaussian white noise, which are assumed to have noise-free constraints and to be uniformly of DAE-index 1. We first provide a rigorous mathematical foundation of the existence and uniqueness of strong solutions. Our theory is based upon the theory of stochastic differential equations (SDEs) and the theory of differential-algebraic equations (DAEs), to each of which our problem reduces on making appropriate simplifications. We then consider discretization methods; implicit methods are necessary because of the differential-algebraic structure, and we consider adaptations of such methods used for SDEs. The consequences of an inexact solution of the implicit equations, roundoff and truncation errors, are analysed by means of the mean-square numerical stability of general drift-implicit discretization schemes for SDEs. We prove that the convergence properties of our drift-implicit Euler scheme, split-step backward Euler scheme, trapezoidal scheme and drift-implicit Milstein scheme carry over from the corresponding properties of these methods applied to SDEs. Finally, we show how the theory applies to the transient noise simulation of electronic circuits.