15.070J | Fall 2013 | Graduate

Advanced Stochastic Processes


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

  1. Review of basic probabilistic concepts; Metric spaces and topology; Topology in ℝd and C([0; T]); Probability on metric spaces.

  2. Large deviations theory.   

    1. Introduction to large deviations; Calculus of large deviations.
    2. Cramer’s theorem, Gartner-Ellis theorem, Sanov’s theorem.
    3. Applications of large deviations methods to queueing systems and to rare event simulations.
  3. Brownian motion theory, martingale theory, Ito calculus.   

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

    1. Probability on metric spaces; Weak convergence of probability measures; Portmentau theorem.
    2. Construction of a Brownian motion; Functional Central Limit Theorem.
    3. 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.


1 Metric spaces and topology  
2 Large deviations for i.i.d. random variables  

Large deviations theory

Cramér’s theorem

4 Applications of the large deviation technique HW 1 due

Extension of LD to ℝd and dependent process

Gärtner-Ellis theorem

6 Introduction to Brownian motion  

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  

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  

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  

Functional law of large numbers

Construction of the Wiener measure


Skorokhod mapping theorem

Reflected Brownian motion

Final Exam

Course Info

Learning Resource Types
Problem Sets with Solutions
Exams with Solutions
Lecture Notes
Instructor Insights