15.070J | Fall 2013 | Graduate

Advanced Stochastic Processes


Reading Materials

The primary reading materials are the course notes prepared by the instructor and distributed before each class.

You will also find exercises in the course notes. The simpler problems are for you to solve while reviewing the notes materials and you are not graded on them. The more difficult ones will be part of the homework assignments. You do not have to work on them before the homework is due!


  • [Billingsley] = Billingsley, Patrick. Weak Convergence of Measures: Applications in Probability. Society for Industrial and Applied Mathematics, 1987. ISBN: 9780898711769. [Preview with Google Books]
  • [Chen and Yao] = Chen, H., and D. Yao. Fundamentals of Queueing Networks: Performance, Asymptotics and Optimization. Springer-Verlag, 2001. ISBN: 9780387951669. [Preview with Google Books]
  • [Dembo and Zeitouni] = Dembo, Amir, and Ofer, Zeitouni. Large Deviations Techniques and Applications. 2nd ed. Springer, 2009. ISBN: 9783642033100. [Preview with Google Books]
  • [Durrett] = Durrett, Rick. Probability: Theory and Examples. 4th ed. Cambridge University Press, 2010. ISBN: 9780521765398. [Preview with Google Books]
  • [Karatzas and Shreve] = Karatzas, Ioannis and Steven, Shreve. Brownian Motion and Stochastic Calculus. 2nd ed. Springer-Verlag, 1991. ISBN: 9780387976556. [Preview with Google Books]
  • [Øksendal] = Øksendal, B. Stochastic Differential Equations: An Introduction with Applications. Springer, 2010. ISBN: 9783540047582. [Preview with Google books]
  • [Resnick] = Resnick, Sydney. Adventures in Stochastic Processes. 1st ed. Birkhauser Verlag, 1992. ISBN: 9780817635916. [Preview with Google Books]
  • [Shwartz and Weiss] = Shwartz, Adam, and Alan Weiss. Large Deviations for Performance Analysis: Queues, Communication and Computing. Chapman and Hall/CRC, 1995. ISBN: 9780412063114. [Preview with Google Books]

Readings by Class Session

1 Metric spaces and topology [Billingsley]: Appendix M1-M10.
2 Large deviations for i.i.d. random variables

[Shwartz and Weiss]: Chapter 0. This is non-technical introduction to the field which describes motivation and various applications of the large deviations theory.

[Dembo and Zeitouni]: Chapter 2.2.


Large deviations theory

Cramér’s theorem

4 Applications of the large deviation technique  

Extension of LD to ℝd and dependent process

Gärtner-Ellis theorem

6 Introduction to Brownian motion

[Resnick]: Sections 6.1, and 6.4 from chapter 6.

[Durrett]: Section 7.1.

[Billingsley]: Chapter 8.


The reflection principle

The distribution of the maximum

Brownian motion with drift

[Resnick]: Sections 6.5, and 6.8 from chapter 6.

[Durrett]: Sections 7.3, and 7.4.

[Billingsley]: Section 9.

8 Quadratic variation property of Brownian motion [Resnick]: Sections 6.11, and 6.12 from chapter 6.
9 Conditional expectations, filtration and martingales [Durrett]: Section 4.1, and 4.2.
10 Martingales and stopping times I [Durrett]: Chapter 4.

Martingales and stopping times II

Martingale convergence theorem

[Durrett]: Chapter 4.

Grimmett, Geoffrey R., and David R. Stirzaker. Section 7.8 in Probability and Random Processes. 3rd ed. Oxford University Press, 2001. ISBN: 9780198572220.

12 Martingale concentration inequalities and applications  
13 Concentration inequalities and applications  
14 Introduction to Ito calculus [Karatzas and Shreve]: Chapter I.
15 Ito integral for simple processes

[Karatzas and Shreve]

[Øksendal]: Chapter III.

Mid-Term Exam
16 Definition and properties of Ito integral

[Karatzas and Shreve]

[Øksendal]: Chapter III.


Ito process

Ito formula

[Øksendal]: Chapter IV.
18 Integration with respect to martingales [Øksendal]: Chapters III, IV, and VIII
19 Applications of Ito calculus to financial economics Duffie, Darrell. Dynamic Asset Pricing Theory. Princeton University Press, 2001. ISBN: 9780691090221. [Preview with Google Books]
20 Introduction to the theory of weak convergence [Billingsley]: Chapter 1. Section 2.
21 Functional law of large numbers Construction of the Wiener measure [Billingsley]: Chapter 2. Section 8.

Skorokhod mapping theorem

Reflected Brownian motion

[Chen and Yao]: Chapter 6.
Final Exam

Supplemental references

Course Info

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