Introduction

This course introduces various crucial notions concerning discrete and continuous time random functions. It has to be seen as a continuation of MA 538.

We begin by a quick review of conditional expectation. Then we will introduce the notion of martingale, which is a class of stochastic processes arising in the description of fair games. We will study the convergence properties of this kind of object, when the time index is discrete. The next topic to be covered concerns ergodic theorems, which can be seen as another tool allowing to get limit theorems for sequences of random variables. Eventually we construct and analyze the most important continuous time stochastic process, namely Brownian motion. According to time, we will also cover some basic notions of stochastic calculus such as Itô's formula.

We begin by a quick review of conditional expectation. Then we will introduce the notion of martingale, which is a class of stochastic processes arising in the description of fair games. We will study the convergence properties of this kind of object, when the time index is discrete. The next topic to be covered concerns ergodic theorems, which can be seen as another tool allowing to get limit theorems for sequences of random variables. Eventually we construct and analyze the most important continuous time stochastic process, namely Brownian motion. According to time, we will also cover some basic notions of stochastic calculus such as Itô's formula.

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Bibliography

- R. Durrett: Probability: theory and examples. Second edition. Duxbury Press, 1996.
- D. Williams: Probability with martingales. Cambridge University Press, 1991.
- R. Durrett: Stochastic calculus. A practical introduction. Probability and Stochastics Series. CRC Press, 1996.
- I. Karatzas, S. Shreve: Brownian motion and stochastic calculus. Second edition. Graduate Texts in Mathematics, 113. Springer-Verlag, 1991.

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Homework

We will follow a

The assignment is below.

**problem list**. Homeworks are usually due on tuesday after class.The assignment is below.

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Problems |
Due |

1-2-4-5-6-7-9 | 8/29 |

10-11-12-13-14 | 9/5 |

15-16-17 | 9/12 |

18-20-21-22-23 | 9/19 |

25-26-27 | 9/26 |

28-29-42-43 | 10/3 |

45-46-47-48 | 10/12 |

50-51-52 | 10/17 |

53-54-55-56 | 10/24 |

57-58-59-60 | 11/14 |

61-62-63-64-65 | 11/21 |

66-67-68-69 | 11/28 |

71-72-73-74-75-76 | 12/5 |

85-86-87-88 | 12/12 |

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There will also be some **reading projects**specified in the link.
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Office hours

2:30-4:00 on Thursday, in Math 434.

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Slides

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