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