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19208001 Stochastik III

Winter Term 2020/2021

Lecture: Prof. Dr. Nicolas Perkowski 
Exercise: Dr. Immanuel Zachhuber


Time and place

  •  Lecture: Mondays, 12--14h, online, and 
                    Wednesdays, 10--12h, online.

  • Exercise Session: Wednesdays, 12--14h, online.
  • Final Exam:  to be announced in due course

Requirements: Prerequisites are Analysis I-III and Stochastics I and II. Functional analysis is helpful but not required.

Assessment

To receive credits fo the course you need to

  • actively participate in the exercise session 
  • work on and successfully solve the weekly exercises 
  • pass the final exam (see above)

Exercises 

Problem sets will be put online every Wednesday and can be found under Assignements in the MyCampus/Whiteboard portal. Solutions (in pairs!) are due on Wednesday of the following week  –  please submit either by uploading the solutions via Whiteboard or by handing them in via email.

Course Overview/ Content:

Announcement video

Stochastic analysis is the study of stochastic processes that evolve in continuous time. We will treat the following subjects, among others:
Gaussian processes; Brownian motion, construction and properties; filtrations and stopping times; continuous time martingales; continuous semimartingales; quadratic variation; stochastic integration; Itô’s formula; Girsanov’s theorem and change of measure; time change; martingale representation; stochastic differential equations and diffusion processes, connections with partial differential equations.

References

  • Jean-François Le Gall: Brownian motion, martingales, and stochastic calculus. Springer, 2016.
  • Ioannis Karatzas and Steven E. Shreve: Brownian motion and stochastic calculus. Springer, 1988.
  • Daniel Revuz and Marc Yor: Continuous martingales and Brownian motion. Springer, 3rd edition, 1999.
  • Achim Klenke: Probability Theory - A Comprehensive Course. Springer, 2008.
  • Peter Mörters and Yuval Peres: Brownian motion. Cambridge University Press, 2010.
  • In the Whiteboard system there will also be lecture notes with additional references.