Year: 2026
Type: dissertation
Abstract: This thesis studies diffusion models from the distributional and pathwise viewpoints. We first describe discrete diffusion models and their continuous-time formulation through stochastic differential equations. The reverse-time SDE depends on the score of the evolving density, which is estimated using denoising score matching. A variance-preserving implementation is then applied to MNIST images of the number 6, where a neural network estimates the added noise and produces recognizable samples. We then study the solution of an SDE as an Itô map that depends on its starting state and Brownian path. The backward flow is introduced as the inverse of this pathwise map. Numerical examples for the OU, variance-preserving, and CIR processes show that the numerical backward equations recover the initial state up to discretization error when the same Brownian increments are used. For the CIR process, the Lamperti transformation considerably improves the reconstruction error. These results show the relation between the distributional and pathwise viewpoints and how they answer different questions.
Author: Brahmi, Hichem
Institution: Concordia University
Language: English
Subfield: Finance
Domain: Social Sciences
Open Access status: green
OpenAlex ID: W7221497243