On Explicit Stochastic Differential Algebraic Equations
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.
Keywords
Cite
@article{arxiv.2102.00605,
title = {On Explicit Stochastic Differential Algebraic Equations},
author = {Sumit Suthar and Soumyendu Raha},
journal= {arXiv preprint arXiv:2102.00605},
year = {2023}
}
Comments
22 pages, 5 figures. In this version of preprint, the definition of the index of SDAE has been modified to make it consistent with our work on SDAEs on smooth manifolds (available at arXiv:2212.13223). The statement of sufficient condition has been modified to address a small error in the previous version. The new statement does not assume the function $g$ to be differentiable