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;



    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


    Notes



    PROBLEM SHEETS


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