## Course Meeting Times

Lectures: 2 sessions / week, 1.5 hours / session

## Prerequisites

*6.431 Applied Probability*, *15.085J Fundamentals of Probability*, or *18.100 Real Analysis (18.100A, 18.100B, or 18.100C)*.

## 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.

## Lecture Topics Overview

- Review of basic probabilistic concepts; Metric spaces and topology; Topology in ℝ
^{d} and *C*([0; T]); Probability on metric spaces. - Large deviations theory.
- Introduction to large deviations; Calculus of large deviations.
- Cramer's theorem, Gartner-Ellis theorem, Sanov's theorem.
- Applications of large deviations methods to queueing systems and to rare event simulations.

- Brownian motion theory, martingale theory, Ito calculus.
- Intro and basic properties of Brownian motion; Reflection principle, quadratic variation.
- Filtration theory, martingales, stopping theory and martingale convergence theorem.
- Concentration inequality for martingales; Applications to the theory of random graphs.
- Stochastic integration and Ito calculus; Applications to finance. Black-Scholes formula.

- Weak convergence theory and applications.
- Probability on metric spaces; Weak convergence of probability measures; Portmentau theorem.
- Construction of a Brownian motion; Functional Central Limit Theorem.
- Applications to the heavy traffic theory of queueing systems.

## Grading

Your grade is based on the in-class midterm exam, take home final exam, and homework problem sets.

## Calendar

Course calendar. LEC # | TOPICS | KEY DATES |

1 | Metric spaces and topology | |

2 | Large deviations for i.i.d. random variables | |

3 | Large deviations theory Cramér's theorem | |

4 | Applications of the large deviation technique | HW 1 due |

5 | Extension of LD to ℝ^{d} and dependent process Gärtner-Ellis theorem | |

6 | Introduction to Brownian motion | |

7 | The reflection principle The distribution of the maximum Brownian motion with drift | |

8 | Quadratic variation property of Brownian motion | HW 2 due |

9 | Conditional expectations, filtration and martingales | |

10 | Martingales and stopping times I | |

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

12 | Martingale concentration inequalities and applications | |

13 | Concentration inequalities and applications | HW 3 due |

14 | Introduction to Ito calculus | |

15 | Ito integral for simple processes | |

**Midterm Exam** |

16 | Definition and properties of Ito integral | |

17 | Ito process Ito formula | HW 4 due |

18 | Integration with respect to martingales | |

19 | Applications of Ito calculus to financial economics | |

20 | Introduction to the theory of weak convergence | |

21 | Functional law of large numbers Construction of the Wiener measure | |

22 | Skorokhod mapping theorem Reflected Brownian motion | |

**Final Exam** |