Constrained Stochastic Differential Equations on Smooth Manifolds.

Abstract

Dynamical systems with uncertain fluctuations are usually modelled using Stochastic Differential Equations (SDEs). Due to operation and performance related conditions, these equations may also need to satisfy the constraint equations. Often the constraint equations are ``algebraic". Such constraint equations along with the given SDE form a system of Stochastic Differential-Algebraic Equations (SDAEs).

The main objective of this thesis is to consider these equations on smooth manifolds. However, we first consider SDAEs on Euclidean spaces to understand these equations locally. A sufficient condition for the existence and uniqueness of the solution is obtained for SDAEs on Euclidean spaces. We also give necessary condition for the existence of the solution. Based on the necessary condition, there exists a class of SDAEs for which there is no solution. Since all SDAEs are not solvable, we present methods and algorithms to find approximate solution of the given SDAE.

In order to extend this work to smooth manifolds, we consider second order stochastic differential geometry to construct Schwartz morphism to represent SDEs with drift that are driven by p-dimensional Wiener process. We show that it is possible to construct such Schwartz morphisms using what we call as \textit{diffusion generators}. We demonstrate that diffusion generator can be constructed using flow of second order differential equations, in particular using regular Lagrangians. The results obtained for SDAEs on Euclidean spaces are extended to SDAEs on smooth manifolds using the framework of diffusion generators. We show that the results obtained for SDAEs on Euclidean spaces translate to the manifold setting with minimal modifications. We have derived Ito-Wentzell's formula on manifolds in the framework of diffusion generators to obtain approximate bounded solution with unit probability. Another type of approximate solution is bounded solution such that the probability of explosion is bounded by $\alpha<1$. We present algorithms to compute approximate solutions of both type. This has been demonstrated with an example of SDAE on a sphere.