# Wintersemester/winter term 2019/20

## Probability with martingales

Modul "Wahrscheinlichkeitstheorie mit Martingalen" (Ma-WTHM)

 Umfang / contact: 3+1 Ort / venue: Willersbau, Zellescher Weg 12 -14 Zeit / time: Mo / Mon 5. DS (14:50-16:20) , WIL C 204 Di / Tue 5. DS (14:50-16:20) , WIL C 204 Übungen / tutorials: während der VL / part of the lecture Beginn / begin: 1 Semesterwoche / 1st week of term Niveau / level : Master / MSc Unterrichtssprache / language: german, english on request Prüfung / exam: oral exam (group examination, group size = 1, duration approx. 20 min) Time: will be set by the examination office Venue: WIL B 319 Registration: with Ms. Schreiter (exam office) Exam language: German (English on request) Prüfungsamt / exam office Aktuelles

• In this course we will continue our study of discrete-time martingales which was started in the BSc module "Stochastik" - we will mainly discusse uniform integrability, L_1-convergence, maximal inequalities - and give a brief introduction to continuous-time martingale theory (path regularization). Then we turn to Gaussian laws and discuss multivariate Gaussian random variables. Finally we begin the study of Brownian motion
• This lecture is the basis for further studies in the direction of stochastic analysis and advanced mathematical finance.
• Prerequisites: measure-theoretic probability theory (e.g. as tought in our BSc course STOCH) and basic knowledge of discrete-time martingales (e.g. as tought in the BSc course STOCH).
• Literature: I will not follow any particular text, but the following books contain (more than) the material covered in this lecture course:
1. Schilling: Martingale & Prozesse. De Gruyter, Berlin 2018. ISBN: 978-3-11-035067-8
2. Schilling: Wahrscheinlichkeit. De Gruyter, Berlin 2017. ISBN: 978-3-11-035065-4
3. Williams: Probability with Martingales. Cambridge University Press. Cambridge 1992. ISBN: 0-521-40605-6
4. Schilling & Partzsch: Brownian Motion. An introduction to the theory of stochastic processes. De Gruyter, Berlin 2014 (2nd edn). ISBN: 978-3-11-030729-0
5. Revuz & Yor: Continuous Martingales and Brownian Motion. Springer, Berlin 2005 (3rd edn). ISBN: 978-3540643258
6. Rogers & Williams: Diffusions, Markov Processes and Martingales. Vol. 1. Cambridge University Press, Cambridge 2000 (2nd edn). ISBN 978-0521775946

Stand:
Autor: René Schilling