Stochastic Differential Equations and Diffusion Processes

SS 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)


Course in the Moodle system: link



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) (link).
    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 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


    • LECTURE NOTES




    PROBLEM SHEETS