Stochastic Processes at CEU -- winter semester 2013/14

HOMEWORKS here.

No. of Credits: 3
No. of ECTS credits: 6
Time Period of the course: Winter Semester
Prerequisites: Probability 1
Course Level: PhD
Syllabus: will be here.
Classes: Wednesdays from 12:00 in room 301.
Course Coordinator: Imre Péter Tóth.

THE FINAL EXAM will be an oral exam on Thursday, the 24th of April 2014, starting at 9:00. LIST OF QUESTIONS HERE.

Topics actually covered:
• Finite dimensional distributions and construction of stochastic processes. Kolmogorov extension theorem.
• Classification of stochastic processes, examples. Poisson process, Wiener process.
• Markov property, finite Markov chains. Mixing and ergodicity of Markov chains.
• Countable Markov chains. Transience, recurrence, positive recurrence. Random walks on Z^d, Pólya's theorem.
• Random walks on Z. The reflection principle and applications. Distribution of the maximum, arcsine law, return times, local time.
• Discrete time martingales. Discrete time stochastic integration. Discrete Black-Scholes formula.
• Brownian motion, Wiener process. Regularity properties, scaling properties.
• Markov chains in continuous time. Infinitesimal generator, exponential semigroup. Examples.
• Filtrations and stopping times. Markov processes and martingales in continuous time.
• Itô's stochastic integral. Motivation, definition, basic properties.
• Itô processes and the Itô formula. Examples of stochastic differential equations.
• Itô representation theorem, Martingale representation theorem.
Suggested literature (References):
1. Specific chapters to some of the lectures:
• For the reflection principle in 1D random waks: http://www.math.bme.hu/~balint/oktatas/hetnet/jegyzet/ltp3_wbh_slides.pdf
• For random walks, including reflection principle: W. Feller: An Introduction to Probability Theory and Its Applications, Vol. 1, Chapter 3
• For the Itô integral, Itô formula, Itô representation theorem, Martingale representation theorem: B. Oksendal: Stochastic Differential Equations: An Introduction with Applications. Fifth Edition. Springer, 2002. Chapters 3 and 4, not including
• pages 27-28,
• Section 3.3,
• Lemma 4.3.1 and the proof of Lemma 4.3.2.
2. General:
• R. Durrett: Probability. Theory and Examples. 4th edition, Cambridge University Press, 2010.
• R. Durrett: Essentials of Stochastic Processes. 2nd edition, Springer, 2012.
• S: Karlin, H.M. Taylor: A First Course in Stochastic Processes. 2nd edition, Academic Press 1975.
• S: Karlin, H.M. Taylor: A Second Course in Stochastic Processes. Academic Press 1981.
• R. Durrett: Stochastic Calculus: A practical Introduction. CRC Press, 1996.
• F.C. Klebaner: Introduction to stochastic calculus with applications. 2nd edition, Imperial College Press, 2005.
• K.L. Chung, R.J. Williams: Introduction to stochastic integration. 2nd edition, Birkhauser 1990.
• D. Revuz, M. Yor: Continuous Martingales and Brownian Motion. 3rd edition, Springer 1999.