LEC # | TOPICS | LECTURE NOTES |
---|---|---|

1 | Metric spaces and topology | Lecture 1: Metric spaces (PDF) |

2 | Large deviations for i.i.d. random variables | Lecture 2: Large deviations technique (PDF) |

3 | Large deviations theory Cramér's theorem | Lecture 3: Cramér's theorem (PDF) |

4 | Applications of the large deviations technique | Lecture 4: Applications of large deviations (PDF) |

5 | Extension of LD to ℝ Gärtner-Ellis theorem | Lecture 5: LD in many dimensions and Markov chains (PDF) |

6 | Introduction to Brownian motion | Lecture 6: Intro Brownian motion (PDF) |

7 | The reflection principle The distribution of the maximum Brownian motion with drift | Lecture 7: Brownian motion (PDF) |

8 | Quadratic variation property of Brownian motion | Lecture 8: Quadratic variation (PDF) |

9 | Conditional expectations, filtration and martingales | Lecture 9: Filtration and martingales (PDF) |

10 | Martingales and stopping times I | Lecture 10: Martingales I (PDF) |

11 | Martingales and stopping times II Martingale convergence theorem | |

12 | Martingale concentration inequalities and applications | Lecture 12: Martigales concentration inequality (PDF) |

13 | Concentration inequalities and applications | Lecture 13: Talagrand's concentration inequality (PDF) |

14 | Introduction to Ito calculus | Lecture 14: Ito calculus (PDF) |

15 | Ito integral for simple processes | Lecture 15: Ito construction (PDF) |

Midterm Exam | ||

16 | Definition and properties of Ito integral | Lecture 16: Ito integral (PDF) |

17 | Ito process Ito formula | Lecture 17: Ito process and formula (PDF) |

18 | Integration with respect to martingales | Notes unavailable |

19 | Applications of Ito calculus to financial economics | Lecture 19: Ito applications (PDF) |

20 | Introduction to the theory of weak convergence | Lecture 20: Weak convergence (PDF) |

21 | Functional law of large numbers Construction of the Wiener measure | Lecture 21: Tightness of measures (PDF) |

22 | Skorokhod mapping theorem Reflected Brownian motion | Lecture 22: Reflected Brownian motion (PDF) |

Final Exam |