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.

- 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.

- 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.

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. |