Reflected stochastic differential equations (RSDEs) are a class of stochastic processes that, loosely speaking, behave like stochastic differential equations (SDEs) in the interior of some bounded domain $\Omega \subseteq \mathbb{R}^d$ while being reflected at the boundary $\partial\Omega$ of $\Omega$, effectively confining them to the compact set $\overline{\Omega}$. Since SDEs are prevalent both in the modelling of real-world phenomena and in machine learning, it is natural to consider how such models and methods might be expanded through the use of RSDEs. In this dissertation, we consider some of the applications of RSDEs, where we first show that these provide novel ways of performing stochastic control, and later that they perform well in certain machine learning methods, where a bounded state space is natural.
In the first article, we consider how the reflection set $\Omega$ can provide a form of control over a stochastic process. In particular, we formulate an intuitive objective function based on the trajectory of an RSDE that expresses the goal of confining a particle to be near the origin without it hitting the boundary of the confining set too often. We then derive a closed form expression of this objective function using probabilistic analysis, and show that this cost function can be optimised both numerically and in a data-driven fashion.
In the second article, inspired by the use of SDEs in generative AI, we show that by replacing the underlying SDEs with RSDEs in these models, we obtain a more natural framework, since the data being generated is often itself bounded. Using spectral decomposition and rigorous neural network constructions, we show that the samples generated converge to the target distribution in total variation at a minimax optimal rate when assuming Sobolev smoothness of the target density.
Finally, in the third article, we continue in the framework of the second, but now considering a type of manifold hypothesis, where the target distribution is concentrated on an affine plane of much lower dimension than the ambient space. Using stochastic analysis, we derive an explicit solution to the involved RSDE, which we use to effectively approximate the target distribution and the affine plane on which it is supported, ultimately yielding near minimax optimal rates in Wasserstein-1-distance under assumptions of Sobolev smoothness.
