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.

Grading rules:
There will be three homework assignments -- roughly once in every four weeks -- worth 20% of the total score. A midterm exam focused on problem solving will be worth 30%, while the final exam, focused on theory will be worth 50%.

Requirements for "audit":
regular participation in class and a short oral account of the concepts and phenomena learned.

Just for the record: Planned schedule, far from what was actually covered:

week #	when	topic	remark
week 1	2014.01.15	Introduction, overview. Finite dimensional distributions and construction of stochastic processes. Kolmogorov extension theorem.
week 2	2014.01.22	Classification of stochastic processes, examples. Poisson process, Wiener process. Markov property, finite Markov chains.
week 3	2014.01.29	Mixing and ergodicity of Markov chains. Countable Markov chains. Transience, recurrence, positive recurrence.	class delayed until spring
week 4	2014.02.05	Random walks on Z. The reflection principle and applications. Distribution of the maximum, arcsine law, return times, local time. Random walks on Z^d.
week 5	2014.02.12	Renewal processes, renewal equations.
week 6	2014.02.19	Discrete time martingales. Branching processes. Barabási-Albert graph model, preferential attachment.
week 7	2014.02.26	Discrete time stochastic integration. Discrete Black-Scholes formula.
week 8	2014.03.05	Brownian motion, Wiener process. Construction(s), regularity properties, scaling properties.
week 9	2014.03.12	Markov chains in continuous time. Infinitesimal generator, exponential semigroup. Examples.
week 10	2014.03.19	Filtrations and stopping times. Markov processes and martingales in continuous time.
week 11	2014.03.26	Ito's stochastic integral. Motivation, definition, basic properties.
week 12	2014.04.02	Stochastic differential equations. Definition, examples.