All readings will be listed here, organized by topic. Dates indicate the
class by which you should have read the indicated material. Some readings
may be announced here before they are announced in class, in case you want to
read ahead. Sometimes, particularly if I am traveling, you may find some
references posted at the library class reserves before they are linked here.

If a reading does not have a date, you don't have to worry about it yet.

Assigned Readings

- Kramer & Majda, "Fundamentals of Probability Theory" (PDF) (09/04/07)
- Grigoriu,
Stochastic Calculus, Secs. 2,1--2,3 (PDF) (09/07/07)- Grigoriu,
Stochastic Calculus, Sec. 2.7 (PDF) (09/07/07)- Grigoriu,
Stochastic Calculus, Secs. 2.3.5, 2.3.6 (PDF) (09/11/07)- Grigoriu,
Stochastic Calculus, Secs. 2.4-2.5 (PDF) (09/11/07)- Grigoriu,
Stochastic Calculus, Sec. 2.10 (PDF) (09/18/07)- Kloeden & Platen,
Numerical Simulation of Stochastic Differential Equations, Sec. 1.3 (PDF) (09/25/07)- Grigoriu,
Stochastic Calculus, Sec. 2.11 (PDF) (10/12/07)- Bertsekas & Tsitsiklis,
Introduction to Probability, Sec. 3.2 (PDF) (10/12/07)- Bertsekas & Tsitsiklis,
Introduction to Probability, Sec. 3.6 (PDF) (10/12/07)- deGroot & Schervish,
Probability and Statistics, Sec. 3.9 (PDF) (10/12/07)

Optional Readings

- Billingsley,
Probability and Measure, Sections 2 and 3 (PDF)

- Introduction to technical and mathematical aspects of measure theoretic foundation of probability theory
- Grigoriu,
Stochastic Calculus, Sec. 2.9 (PDF)

- Technical discussion of Radon-Nikodym derivative
- P. L'Ecuyer, ``Pseudorandom Number Generators'', in
Encyclopedia of Quantitative Finance, Rama Cont, Ed., in volumeSimulation Methods in Financial Engineering, Eckhart Platen and Peter Jaeckel Eds., Wiley, forthcoming (PDF)

- Survey of pseudorandom number generators

Moment Generating Functions and Characteristic Functions

Assigned Readings

- Grigoriu,
*Stochastic Calculus*, Sec. 2.10.3.1 (PDF) (10/19/07) - Bertsekas & Tsitsiklis,
*Introduction to Probability*, Secs. 4.1-4.2 (11/09/07)- or any textbook discussion of the use of moment generating functions and their relation to sums of independent random variables

Random Walks

Assigned Readings

- Feller,
An Introduction to Probability Theory and its Applications, Ch. III (PDF) (11/13/07)

- Sections 5 and 8 are optional, and pertain to further clever arguments involving calculating path properties of symmetric random walks
- Resnick,
Adventures in Stochastic Processes,Sec. 1.6 (PDF) (11/13/07)

Optional Readings

- Resnick,
Adventures in Stochastic Processes,Sec. 1.8 (PDF) (11/13/07)

- Rigorous discussion of splitting random walks into sums of independent segments

Conditional Probability and Expectation

Assigned Readings

- Bertsekas & Tsitsiklis,
Introduction to Probability, Secs. 1.3, 1.4, 2.6 (11/20/07)

- or other elementary textbook discussion of conditional probability
Bertsekas & Tsitsiklis,

, Sec 3.4 (PDF) (11/27/07)Introduction to Probability- Bertsekas & Tsitsiklis,
, Sec 3.5 (12/04/07)Introduction to Probability

- or other elementary textbook discussion of conditional probability and expectation with respect to realizations of continuous random variables
- Grigoriu,
Stochastic Calculus, Sec. 2.17 (PDF) (12/04/07)- Bertsekas & Tsitsiklis,
, Secs. 4.3, 4.4 (PDF) (12/07/07)Introduction to Probability

Optional Readings

- K. Kiyono, Zbigniew R. Struzik, and Yoshiharu Yamamoto, "Criticality and Phase Transition in Stock-Price Fluctuations."
Phys. Rev. Lett.96, 068701 (2006).

- Example of probability model for stock price fluctuations which takes the form of a randomized probability distribution: Gaussian conditioned on the volatility, and the volatility has lognormal distribution

Central Limit Theorem

Assigned Readings

- deGroot & Schervish,
Probability and Statistics, Sec. 5.7 (PDF) (12/12//07)

Optional Readings

- Grigoriu,
Stochastic Calculus, Secs. 2.12 & 2.13 (PDF)

- Mathematical foundations for rigorous estimates and convergence studies in probability theory
- Sinai,
Probability Theory, Lect. 15 (PDF)

- Rigorous proof of central limit theorem

Multivariate Gaussian Random Variables

Assigned Readings