On Explicit Stochastic Differential Algebraic Equations

TL;DR

This paper investigates the mathematical properties of Stochastic Differential-Algebraic Equations (SDAEs), establishing conditions for existence and uniqueness of solutions, and proposing approximation methods for high index cases, supported by examples and numerical analysis.

Contribution

It combines DAE theory with SDEs to analyze SDAEs, providing new existence, uniqueness conditions, and approximation techniques for high index equations.

Findings

Established sufficient conditions for solution existence and uniqueness.

Identified classes of high index SDAEs with no solutions.

Developed approximation methods for high index SDAEs.

Abstract

Dynamical systems that are subject to continuous uncertain fluctuations can be modelled using Stochastic Differential Equations (SDEs). Controlling such system results in solving path constrained SDEs. Broadly, these problems fall under the category of Stochastic Differential-Algebraic Equations (SDAEs). In this article, the focus is on combining ideas from the local theory of Differential-Algebraic Equations with that of Stochastic Differential Equations. The question of existence and uniqueness of the solution for SDAEs is addressed by using contraction mapping theorem in an appropriate Banach space to arrive at a sufficient condition. From the geometric point of view, a necessary condition is derived for the existence of the solution. It is observed that there exists a class of completely high index SDAEs for which there is no solution. Hence, techniques to find approximate solution of completely high index equations are presented. The techniques are illustrated with examples and numerical computations.