Professur für Wahrscheinlichkeitstheorie

Sommersemester / Summer term 2018

Stochastic Calculus

Modul "Stochastic Calculus" (Ma-STOCHC)

Umfang / contact: 3+1
Ort / venue: Willersbau, Zellescher Weg 12 -14
Zeit / time: Mi / Wed 1. DS (07:30-09:00)    WIL C204
Fr / Fri    1. DS (07:30-09:00)    WIL C204
Übungen / tutorials: während der VL / part of the lecture
Beginn / begin: 1 Semesterwoche / 1st week of term
Niveau / level : Master / MSc
Unterrichtssprache /
Prüfung / exam: oral exam (group examination, group size = 1, duration approx. 20 min)
Registration: with Ms. Schreiter (exam office)
Exam language: either English (oder, auf Wunsch, Deutsch)
Prüfungsamt Aktuelles 

  • In this course we will discuss stochastic integration and stochastic differential equations. We begin with  Brownian motion to explore the scope and problems of stochastic integration, then we discuss the structure of martingales in order to introduce a general stochastic integral. Our aim are SDEs (stochastic differential equations) and some of their qualitative properties.
  • This lecture is the basis for further studies in the direction of stochastic analysis and advanced mathematical finance.  I plan to offer a follow-up course or seminar on Malliavin Calculus (Stochastic Calculus of Variations) or on Lévy Processes (jump processes) in the following winter term.
  • Prerequisites: measure-theoretic probability theory (e.g. as tought in our BSc course STOCH) and basic knowledge of martingales (e.g. as tought in the BSc course STOCHV).  Although this is formally a follow-up to PWM, this lecture can be attended independentliy of PWM provided that you have a basic working knowledge of Brownian motion (e.g. as taught in PWM, or as described in my book on Brownian motion, Sections 2.1, 2.2, 5.1, 5.2 - see below)
    • Literature: The first part of the lecture (BM and SDEs) will be based on my own book
      • Schilling & Partzsch: Brownian Motion. An introduction to the theory of stochastic processes. De Gruyter, Berlin 2014 (2nd edn). ISBN: 978-3-11-030729-0
    • Further Literature:
      1. Borodin: Stochastic Processes. Birkhäuser 2017.
      2. Ikeda & Watanabe: Stochastic Differential Equations and Diffusion Processes. North-Holland 1989 (2nd edn). ISBN: 978-1493307210
      3. Kunita: Stochastic Flows and Stochastic Differential Equations. Cambridge Univ. Press, Cambridge 1990. ISBN: 0-521-35050-6
      4. Revuz & Yor: Continuous Martingales and Brownian Motion. Springer, Berlin 2005 (3rd edn). ISBN: 978-3540643258

Autor: René Schilling