This seminar provides an overview of stochastic differential equations (SDEs) with a focus on their relevance in understanding diffusion models, which are considered state-of-the-art deep generative models. The seminar is scheduled for the first half of the winter term, and participants have the option to attend a follow-up seminar titled Score-based Generative Models for Machine Learning (Master Seminar), which takes place in the second half of the winter term.
The seminar covers a wide range of topics without delving into minute details. Instead, it aims to address the most essential aspects related to the aforementioned generative models. The content of the seminar is structured as follows:

• Review of Differential Equations: The seminar begins with a review of fundamental concepts in differential equations, with a specific emphasis on the initial value problem. It covers key results related to existence, uniqueness, and numerical analysis for integration.
• Mathematical notations and statistical concepts of stochastic processes: We revisit the basic mathematical notations and introduce necessary statistical concepts essential for introducing the Ito integral.
• Ito calculus: Participants will learn about Ito integrals and their main properties. Special attention is given to the Ito isometry. The seminar will include derivations, examples, and applications of Ito integrals.
• Ito Formula: We derive the Ito formula, providing insights into its significance and practical applications.
• Statistics of SDEs: The final part of the seminar focuses on the statistical aspects of SDEs. Topics include the derivation of the Fokker-Planck-Kolmogorov equation, an examination of the markov and martingale properties of SDEs, and the derivation of the equations for the moments of SDEs.

By the end of the seminar, participants will have a better understanding of SDEs with insights in the context of diffusion models. This knowledge can be valuable for those interested in advanced topics in machine learning and mathematical modeling.

• Prerequisites: Basic knowledge in probability theory and statistics
• Registration: Via Müsli. Link
• First (organizational) meeting: Tuesday, 17 October at 11:00 c.t.
• Time and Location: Tuesdays 14:00 c.t. in SR 6

Further information on the seminar will be announced in the first organizational meeting. For any specific question you can contact Daniel Gonzalez, Jonas Cassel.

• Applied stochastic differential equations, Särkkä, Simo and Solin, Arno, Cambridge University Press (2019)
• Stochastic differential equations: an introduction with applications, Oksendal, Bernt, Springer Science & Business Media (2013)
• An introduction to stochastic differential equations, Evans, Lawrence C American Mathematical Soc. (2012)
• Analysis 2. Differential-und Integralrechnung für Funktionen mehrerer reeller Veränderlichen, Rannacher, Rolf Heidelberg University Publishing (2018)