Course Meeting Times
Lectures: 2 sessions / week, 1.5 hours / session
6.431 Applied Probability, 15.085J Fundamentals of Probability, or 18.100 Real Analysis (18.100A, 18.100B, or 18.100C).
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 ℝd 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.
Your grade is based on the in-class midterm exam, take home final exam, and homework problem sets.
|LEC # ||TOPICS ||KEY DATES |
|1 ||Metric spaces and topology || |
|2 ||Large deviations for i.i.d. random variables || |
|3 || |
Large deviations theory
|4 ||Applications of the large deviation technique ||HW 1 due |
|5 || |
Extension of LD to ℝd and dependent process
|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 || |
|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 |