Date of Award

Spring 2026

Language

English

Embargo Period

5-10-2026

Document Type

Master's Thesis

Degree Name

Master of Arts (MA)

College/School/Department

Department of Mathematics and Statistics

Program

Mathematics

First Advisor

Karin Reinhold

Committee Members

Martin Hildebrand

Subject Categories

Statistical Models

Abstract

Hidden Markov Models (HMMs) are a powerful mathematical system used for analyzing time sequences in which the underlying states are unable to be directly observed. This project strives to develop the theory and computation of such models, including deriving the forward and backward variables, the Baum-Welch algorithm, and the Viterbi algorithm. Each of these HMM components are derived and explained to demonstrate the probabilistic principles that allow HMMs to be effective for modeling sequential data.

To demonstrate its practical relevance, the HMM is applied to financial time series. The observable stock market returns tend to be influenced by market regimes that are hidden, such as growth, decline, or high volatility. To handle the continuous nature of stock market returns, Gaussian Mixture Models are incorporated. This also allows the system to capture the complex statistical patterns and identify latent market states, which help provide insight into the regime transitions. This work emphasizes how rigorous mathematical modeling can be applied to real-world systems, by combining theoretical development with a practical analysis, in order to understand and better interpret stock market behavior.

License

Creative Commons Attribution 4.0 International License
This work is licensed under a Creative Commons Attribution 4.0 International License.

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