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, GartnerEllis 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. BlackScholes 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 inclass midterm exam, take home final exam, and homework problem sets.
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 GärtnerEllis 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 