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Dozent

Jun.-Prof. Dr. Zakhar Kabluchko

Ãœ²ú³Ü²Ô²µ²õ±ô±ð¾±³Ù±ð°ù

Dipl.-Math. Stefan Roth
Dipl.-Math. oec. Judith Schmidt

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Zeit und Ort

Vorlesung
Dienstag, 10 - 12 Uhr, H14

Donnerstag, 10 - 12 Uhr, H14

Ãœ²ú³Ü²Ô²µ
Mittwoch, 16 - 18 Uhr, H14

The course can be held in English. 

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Art der Vorlesung

4 Stunden Vorlesung + 2 Stunden Ãœ²ú³Ü²Ô²µ

Leistungspunkte: 9

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Voraussetzungen

Elementare Wahrscheinlichkeitsrechnung und Statistik. Stochastik I ist nützlich, wird aber nicht vorausgesetzt. 

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Zielgruppe

Pflichtmodul ´Úü°ù:

Master Wirtschaftsmathematik

Wahlpflichtmodul ´Úü°ù:

Bachelor Mathematik, Mathematische Biometrie, Wirtschaftsmathematik;
Master Mathematik

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Inhalt

Die Vorlesung gibt eine Einführung in die Theorie der stochastischen Prozesse.  Schwerpunkte der Vorlesung sind:

• Markov-Ketten
• Poisson-Prozess
• Wiener-Prozess (Brownsche Bewegung)
• Martingale
• Unbegrenzt teilbare und stabile Verteilungen
• ³¢Ã©±¹²â-±Ê°ù´Ç³ú±ð²õ²õ±ð
• Stationäre Prozesse in diskreter Zeit

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Vorleistung

50% der Ãœ²ú³Ü²Ô²µspunkte. Zur Anrechnung der erzielten Ãœ²ú³Ü²Ô²µspunkte ist eine Anmeldung mit notwendig.

Um die Vorleistung zu bestehen sind 130 Punkte im SLC notwendig.

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Klausur

The second exam took place on April 2, 2014, in H4/5 at 13:00-15:00.

The inspection of the second exam takes place in room E60 (Helmholtzstr. 18) on Thursday, April 10, at 14-16 o'clock.

The results of the second exam are posted online in the . If you are not registered in the SLC, please write a short email to Judith Olszewski LINK. To determine your grade, please use the following table:

Grade Points

1.0    66-74

1.3    61-65

1.7    57-60

2.0    54-56

2.3    51-53

2.7    48-50

3.0    45-47

3.3    42-44

3.7    39-41

4.0    35-38

5.0    0-34

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Vorlesungsmanuskript

Das Skript wird hier online gestellt. Lecture notes will be posted here. 

• Chapter 1. General theory of stochastic processes: PDF
• Chapter 2. Markov Chains. Part 1: PDF      Part 2: PDF
  Both parts together: PDF
  Chapter 2 of these lecture notes follows closely a book by James Norris: "Markov Chains", Cambridge University Press.
  Parts of this book can be downloaded at the homepage of the author:
  We also recommend the following books on Markov chains:
  1) G. Lawler. Introduction to the theory of stochastic processes.
  2) N. Privault. Understanding Markov chains. 
  Lecture notes of N. Privault related to this book can be downloaded here:
• Chapter 3. Renewal Processes and the Poisson Process:  PDF
  We recommend the following books:
  1) G. Lawler. Introduction to the theory of stochastic processes.
  2) S. Resnick. Advenrutes in stochastic processes.
• Chapter 4. Brownian Motion: PDF
  We recommend the book
  P. Mörters, Y. Peres. Brownian Motion. 
• Chapter 5. Strong Markov Property. Lecture notes will be posted later...
  We recommend the book
  P. Mörters, Y. Peres. Brownian Motion. 
• Chapter 6. Martingales. An excellent book on the topic is
  D. Williams. Probability with martingales. 
• Chapter 7. Levy Processes.
• Chapter 8. Stochastic Integrals. We recommend the books:
  Hui-Hsiung Kuo. Introduction to Stochastic Integration.
  Thomas Mikosch. Elementary Stochastic Calculus with Finance in View.

Very nice lecture notes of Steven Lalley (University of Chicago) can be downloaded here: and here:

The following book is highly recommended: R. Durrett. Probability: Theory and Examples. 

Skript zu Stochastik II von Prof. Schmidt:

Skript zu Stochastik II von Prof. Spodarev:

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Ãœ²ú³Ü²Ô²µsblätter

Exercise sheet No. 1 (Due to: 23rd of October 2013)

Exercise sheet No. 2 (Due to: 30th of October 2013)

Exercise sheet No. 3 (Due to: 5th of November 2013)

Exercise sheet No. 4 (Due to: 13th of November 2013)

Exercise sheet No. 5 (Due to: 20th of November 2013)

Exercise sheet No. 6 (Due to: 27th of November 2013), ·¡°ù²µÃ¤²Ô³ú³Ü²Ô²µ±ð²Ô

Exercise sheet No. 7 (Due to: 4th of December 2013)

Exercise sheet No. 8 (Due to: 11th of December 2013), Hinweis: Änderung in Aufgabe 1

Exercise sheet No. 9 (Due to: 18th of December 2013)

Exercise Sheet No. 10 (Due to: 8th January 2014. Merry Christmas and a Happy New Year!)

Exercise Sheet No. 11 (Due to: 15th January 2014)

Exercise sheet No. 12 (Due to: 22nd January 2014), Hinweis: Korrektur bei Aufgabe 2, (c)

Exercise sheet No. 13 (Due to: 29th January 2014)

Exercise Sheet No. 14 (Due to 5th February 2014)

Exercise Sheet No. 15 (Please don't submit solutions. Solutions to selected problems from this exercise sheet will be discussed on February 12 during the last exercise session). 

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Literatur

• R. Durrett. Essentials of Stochastic Processes, Springer Texts in Statistics.
• R. Durrett. Probability: Theory and Examples, Cambridge University Press.
• P. Mörters, Y. Peres. Brownian Motion. Cambridge University Press.
• G. Kersting, A. Wakolbinger. Zufallsvariable und Stochastische Prozesse. Birkhäuser.
• R. Bass. Stochastic Processes, Cambridge University Press.
• D. Williams. Probability with Martingales, Cambridge University Press.
• J. Lamperti. Probability: A Survey of Mathematical Theory, Wiley.
• G. Grimmett, D. Stirzaker. Probability and Random Processes. Oxford University Press. 
• S. Resnick. Adventures in Stochastic Processes. Birkhäuser.
• G. Lawler. Introduction to Stochastic Processes. Chapman and Hall.
• G. Lawler, L. Coyle. Lectures on Contemporary Probability. American Mathematical Society. 
• N. U. Prabhu. Stochastic Processes: Basic Theory and Its Applications. World Scientific. 
• J. F. C. Kingman. Poisson Processes. Oxford University Press.