# Syllabus

## Description

The class covers the analysis and modeling of stochastic processes. Topics include measure theoretic probability, martingales, filtration, and stopping theorems, elements of large deviations theory, Brownian motion and reflected Brownian motion, stochastic integration and Ito calculus and functional limit theorems. In addition, the class will go over some applications to finance theory, insurance, queueing and inventory models.

## Grading

Grading for this class will be based on the bi-weekly homework assignments, a mid-term and a final exam.

## Calendar

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

1 | Probability Basics: Probability Space, σ-algebras, Probability Measure |

2 | Random Variables and Measurable Functions; Strong Law of Large Numbers (SLLN) |

3 | Large Deviations for i.i.d. Random Variables |

4 | Large Deviations Theory (cont.) (Part 1) Properties of the Distribution Function G (Part 2) |

5 | Brownian Motion; Introduction |

6 | The Reflection Principle; The Distribution of the Maximum; Brownian Motion with Drift |

7 | Quadratic Variation Property of Brownian Motion |

8 | Modes of Convergence and Convergence Theorems |

9 | Conditional Expectations, Filtration and Martingales |

10 | Martingales and Stopping Times |

11 | Martingales and Stopping Times (cont.); Applications |

12 | Introduction to Ito Calculus |

13 | Ito Integral; Properties |

14 | Ito Process; Ito Formula |

15 | Martingale Property of Ito Integral and Girsanov Theorem |

16 | Applications of Ito Calculus to Finance |

17 | Equivalent Martingale Measures |

18 | Probability on Metric Spaces |

19 | σ-fields on Measure Spaces and Weak Convergence |

20 | Functional Strong Law of Large Numbers and Functional Central Limit Theorem |

21 | G/G/1 Queueing Systems and Reflected Brownian Motion (RBM) |

22 | Fluid Model of a G/G/1 Queueing System |

23 | Fluid Model of a G/G/1 Queueing System (cont.) |

24 | G/G/1 in Heavy-traffic; Introduction to Queueing Networks |

25 | Final Notes and Ongoing Research Questions and Resources |