Ph.D. Thesis {Colloquium}: CDS: 09th February 2023 : “Stochastic Differential Equations with Constraints on Smooth Manifolds”.

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). We consider these equations on Euclidean spaces and on smooth manifolds. We address the questions on solvability of these equations and give numerical/analytical methods to solve the equations.

This work is divided into three main parts.

1. In the first part, we consider SDAEs on Euclidean spaces and give sufficient condition for the existence and uniqueness of the solution. We also give necessary condition for the existence of the solution. Based on the necessary condition, we obtain a class of SDAEs for which there is no solution. Since, not every SDAE is solvable, we present methods and algorithms to find approximate solution of the given SDAE.

2. In order to extend this work to smooth manifolds, we use Schwartz’s second order stochastic differential geometry to represent SDEs in a coordinate invariant way. We consider Schwartz morphisms from $\mathbb{R}^{p+1}$ to manifold $M$, such that it morphs the semi-martingale $(t,W_t)\in \mathbb{R}^{p+1}$ to a semi-martingale on $M$. We show that it is possible to construct such Schwartz morphisms using special fiber preserving map from the tangent bundle to the Schwartz’s second order tangent bundle. We call these maps as diffusion generators. We demonstrate that diffusion generator can be constructed using flow of first order and second order differential equations. In particular we show that for manifolds without a connection, it is possible to describe Ito-type SDEs using regular Lagrangians. Based on the diffusion generator approach of describing SDEs on manifolds, we also give Ito-Stratonovich-Schwartz conversion forlumae, and the generalized Ito-type formula on manifolds.

3. Lastly, using the diffusion generator approach to describe SDEs on manifolds, we extend the results of SDAEs on Euclidean spaces to the smooth manifolds setting. We show that all the results translate to the manifold setting with minimal modifications.

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