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.

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. We start this seminar reviewing the most important results on differential equations, putting a special focus on the initial value problem. We review existence and uniqueness results as well as the numerical analysis for integration. We continue the seminar revisiting the basic mathematical notations and statistical concepts needed to introduce the Ito integrals. We derive the most important properties for the Ito calculus, as it is the Ito isometry. Having this setup, we derive the Ito formula and handle some examples and applications. We conclude this seminar studying the statistics of SDEs. We derive the Fokker-Planck-Kolmogorov equation, review the Markov and Martigale Properties of SDEs, and derive general equations for the moments of SDEs.

• Prerequisites: Basic knowledge in probability theory and statistics
• Registration: Via Müsli. Link
• First (organizational) meeting: Kalenderwoche 42. Specific day and time will be announced soon.
• Time and Location: Time and location will be announced soon.

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

• 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)