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) (01/31/11)
- Minier & Peirano, "The PDF Approach to Turbulent Polydispersed Two-Phase Flows,"
Physics Reports352(2001), Sec. 2 (PDF) (01/31/11)- Kramer & Majda, "Conditional Probability and Expectation" (PDF) (02/10/11)
- Bertsekas & Tsitsiklis,
Introduction to Probability Theory, Sec. 3.5 (PDF) (02/10/11)- Grigoriu,
Stochastic Calculus, Sec. 2.11(PDF) (02/14/11)- Oksendal,
Stochastic Differential Equations, Appendix A (PDF) (02/14/11)- Kloeden & Platen,
Numerical Solution of Stochastic Differential Equations, Sec. 1.3 (PDF) (02/17/11)- Bertsekas & Tsitsiklis,
Introduction to Probability Theory, Sec. 4.1 (PDF) (02/24/11)- Grigoriu,
Stochastic Calculus, Sec. 2.17 (PDF) (02/24/11)

Assigned Readings

- Kramer & Majda, "Wiener Process" (PDF) (03/24/11)
- Shreve, Stochastic Calculus for Finance II: Continuous-Time Models, Sec. 3.4 (PDF) (04/17/11)

Optional Readings

- Simon,
Functional Integration & Quantum Physics, Sec. 5 (PDF)

- Discussion of Feynman path integral and other ways to define Brownian motion in a mathematically rigorous manner. Also some resuls on smoothness of stochastic processes.
- Oksendal,
Stochastic Differential Equations, Ch. 2 (PDF)

- Technical graduate-mathematics discussion of Wiener process
- Stroock, "Gaussian Measure on a Hilbert Space" (PDF)

- Notes from a graduate summer school on probability theory describing a direct definition of the Wiener process through a Gaussian probability measure on the function space of continuous functions.
- Kramer, "Brownian Motion" (PDF)

- A brief encyclopedia article on use of Brownian motion in nonlinear science models

Assigned Readings

- Gardiner, Handbook of Stochastic Methods Sec. 4.3 (PDF) (04/18/11)
- Shreve, Stochastic Calculus for Finance II: Continuous-Time Models, Secs. 4.4-4.5 (PDF) (04/18/11)

Optional Readings

- Kupferman, Pavliotis, and Stuart, "Itô versus Stratonovich white-noise limits for systems with inertia

and colored multiplicative noise," Phys. Rev. E. 70, 036120 (2004) (PDF).

- Discussion of how Ito or Stratonovich stochastic differential equations emerge as coarse-graining of a specific model, depending on relative size of two small time scale

- Kloeden & Platen,
Numerical Solution of Stochastic Differential Equations, Secs. 4.3--4.4 (PDF)

- Methods of solving some classes of nonlinear stochastic differential equations

- Shreve, Stochastic Calculus for Finance II: Continuous-Time Models, Secs. 4.7 (PDF)

- discussion of Brownian bridge model

- discussion of local time correction to Ito formula

Assigned Readings

- Higham, "An Algorithmic Introduction to Numerical Simulation of Stochastic Differential Equations,"
SIAM Review43(3), 2001: 525-546 (PDF) (04/25/11)

Optional Readings

- Talay, "Simulation of Stochastic Differential Systems," in Probabilistic Methods in Applied Physics, P. Kree and W. Wedig (eds.), Lecture Notes in Physics 451, Springer-Verlag, 1995 (PDF)

- A practical advanced survey on numerical methods for stochastic differential equations
- Kloeden & Platen,
Numerical Solution of Stochastic Differential Equations, Sec. 9.8 (PDF)

- Stability issues in numerical simulations of stochastic differential equations
- Kloeden & Platen,
Numerical Solution of Stochastic Differential Equations, Sec. 11.1 (PDF)

- Strong first order Runge-Kutta-like (derivative free) method

- Almgren, "Financial Derivatives and Partial Differential Equations,"
*The American Mathematical Monthly***109**(1), 2002: 1-12 (05/09/11) (PDF)

- Minier & Peirano, "The PDF Approach to Turbulent Polydispersed
Two-Phase Flows,"
*Physics Reports***352**(2001), Secs. 2&4 (PDF) (05/09/11) - Kramer, "Fokker-Planck Equation" (PDF) (05/09/11)

- Kramer & Majda, "Diffusion Equation" (PDF)
- Kramer & Majda, "Method of Characteristics" (PDF)
- Risken,
*The Fokker-Planck Equation*, Section 6.6 (PDF)- Summary of methods for obtaining solutions or approximate solutions to Fokker-Planck Equation

- Oksendal,
*Stochastic Differential Equations*, Sections 7A & B (PDF)- Rigorous explanation of Markov and strong Markov properties of solutions to stochastic differential equations