Stochastic Differential Equations and Diffusion Processes
Summer Semester 2021
- Lectures: Tuesday 12:15–13:45, Online, link to Zoom
- Tutorial classes: Wednesday 14:15–15:45, Online, link to Zoom
- Office hours: Monday 17:00–18:00, Online, link to Zoom (different from course link)
- Link to course in the Moodle system
Topics
The course aims to develop applications of stochastic calculus to studying stochastic processes in continuous time. The following topics will be covered
- Stochastic differential equations:
- strong and weak solutions;
- martingale problem and uniqueness;
- Yamada-Watanabe theorem;
- semigroups for diffusion processes;
- strong Markov property of solutions;
- comparison principle.
- Local time:
- local time and Tanaka formula;
- reflected Brownian motion;
- sticky-reflected Brownian motion (time change, non-existence, and non-uniqueness of strong solution);
- stochastic differential equations in a domain;
- Poisson point process of Brownian excursions. Representation of Brownian motion, reflected Brownian motion, and sticky reflected Brownian motion via the Poisson point process of Brownian excursions.
- Conditional diffusions:
- Brownian bridge;
- Doob's transform;
- condition a diffusion to not leave a domain;
- Bessel process and conditioning Brownian motion to stay positive;
- conditioning of independent Brownian motions to coalesce.
Prerequisites
Martingales, Brownian motion, stochastic integral, Ito’s formula etc. The required topics were covered by Prof. Dr. Mathias Trabs in “Stochastic calculus” (WS 2020/21).
Suggested references: Prof. Trabs's lecture notes (will be available soon on his webpage) or e.g. Prof. Eberle’s lecture notes Stochastic Analysis
Literature
- Anton Bovier, “Introduction to stochastic analysis”, Lecture Notes, Bonn, Winter Semester 2017/18
- Alexander S. Cherny and Hans-Juergen Engelbert, “Singular stochastic differential equations”
- Hans-Juergen Engelbert and Goran Peskir, “Stochastic differential equations for sticky Brownian motion”, Stochastics 86 (2014), no. 6, 993–1021
- Nobuyuki Ikeda and Shinzo Watanabe, “Stochastic differential equations and diffusion processes”
- Olav Kallenberg, “Foundations of modern probability”
- Daniel Revuz and Marc Yor, “Continuous martingales and Brownian motion”
- Timo Seppaelaeinen, “Basics of stochastic analysis”, Lecture Notes, 2014
- Shinzo Watanabe, “Ito’s theory of excursion point processes and its developments”, Stochastic Process. Appl. 120 (2010), no. 5, 653–677
- Toshio Yamada and Shinzo Watanabe, “On the uniqueness of solutions of stochastic differential equations”, J. Math. Kyoto Univ. 11 (1971), 155–167