LEC # | TOPICS | READINGS |
---|---|---|

1 | Metric spaces and topology | [Billingsley]: Appendix M1-M10. |

2 | Large deviations for i.i.d. random variables | [Shwartz and Weiss]: Chapter 0. This is non-technical introduction to the field which describes motivation and various applications of the large deviations theory. [Dembo and Zeitouni]: Chapter 2.2. |

3 | Large deviations theory Cramér's theorem | |

4 | Applications of the large deviation technique | |

5 | Extension of LD to ℝ Gärtner-Ellis theorem | |

6 | Introduction to Brownian motion | [Resnick]: Sections 6.1, and 6.4 from chapter 6. [Durrett]: Section 7.1. [Billingsley]: Chapter 8. |

7 | The reflection principle The distribution of the maximum Brownian motion with drift | [Resnick]: Sections 6.5, and 6.8 from chapter 6. [Durrett]: Sections 7.3, and 7.4. [Billingsley]: Section 9. |

8 | Quadratic variation property of Brownian motion | [Resnick]: Sections 6.11, and 6.12 from chapter 6. |

9 | Conditional expectations, filtration and martingales | [Durrett]: Section 4.1, and 4.2. |

10 | Martingales and stopping times I | [Durrett]: Chapter 4. |

11 | Martingales and stopping times II Martingale convergence theorem | [Durrett]: Chapter 4. Grimmett, Geoffrey R., and David R. Stirzaker. Section 7.8 in |

12 | Martingale concentration inequalities and applications | |

13 | Concentration inequalities and applications | |

14 | Introduction to Ito calculus | [Karatzas and Shreve]: Chapter I. |

15 | Ito integral for simple processes | [Karatzas and Shreve] [Øksendal]: Chapter III. |

Mid-Term Exam | ||

16 | Definition and properties of Ito integral | [Karatzas and Shreve] [Øksendal]: Chapter III. |

17 | Ito process Ito formula | [Øksendal]: Chapter IV. |

18 | Integration with respect to martingales | [Øksendal]: Chapters III, IV, and VIII |

19 | Applications of Ito calculus to financial economics | Duffie, Darrell. Dynamic Asset Pricing Theory. Princeton University Press, 2001. ISBN: 9780691090221. [Preview with Google Books] |

20 | Introduction to the theory of weak convergence | [Billingsley]: Chapter 1. Section 2. |

21 | Functional law of large numbers Construction of the Wiener measure | [Billingsley]: Chapter 2. Section 8. |

22 | Skorokhod mapping theorem Reflected Brownian motion | [Chen and Yao]: Chapter 6. |

Final Exam |