AM281 S01 -- Stochastic Differential Equations, Fall 2005


Instructor: Paul Dupuis, pdupuis [at] dam.brown.edu
Lecturers: Luis Carvalho, carvalho [at] dam.brown.edu
Yingda Cheng, ycheng [at] dam.brown.edu
Tom Dean, Thomas_A_Dean [at] brown.edu
Matt Feiszli, mfeisz [at] dam.brown.edu
Jasmine Foo, jfoo [at] dam.brown.edu
Sergey Kushnarev, skushn [at] dam.brown.edu
Michael Lamar, mlamar [at] dam.brown.edu
Kevin Leder, kleder [at] dam.brown.edu
Akil Narayan, anaray [at] dam.brown.edu
Ravi Srinivasan, rav [at] dam.brown.edu
Description: This course develops the theory and some applications of stochastic differential equations. Topics include: stochastic integral with respect to Brownian motion, existence and uniqueness for solutions of SDEs, Markov property of solutions, sample path properties, Girsanov's Theorem, weak existence and uniqueness, and connections with partial differential equations. Possible additional topics include stochastic stability, reflected diffusions, numerical approximation, and stochastic control. Prerequisite: AM 264
Schedule: Thursday 12:00-1:00, Friday 11:00-12:30, conference room in 180 George St.
Textbooks: Numerical Solution of Stochastic Differential Equations (Applications of Mathematics), 1st edition; Peter E. Kloeden, Eckhard Platen; Springer; ISBN 3540540628 [amazon]

Stochastic Differential Equations: An Introduction with Applications (Universitext), 6th edition; Bernt Øksendal; Springer; ISBN 3540047581 [amazon]

Brownian Motion and Stochastic Calculus (Graduate Texts in Mathematics), 2nd edition; Ioannis Karatzas, Steven E. Shreve; Springer; ISBN 0387976558 [amazon]


Homework:
  • HW set 1: Øksendal, #2.14, 2.15, 3.7, 3.10, 3.15, 4.10, 4.12, 4.13, 5.2, 5.9, 5.11, 5.15, 7.9, 7.14. [pdf]
  • HW set 2: Øksendal, #8.11, 8.13, 9.12, 11.6, 11.12. [pdf]
Core topics:
  • Background material--martingales and martingale bounds in continuous time. [pdf]
    (Akil Narayan)
  • Review of properties of Brownian motion. [pdf]
    (Luis Carvalho)
  • Stochastic integral with respect to Brownian motion--construction and properties for continuous integrands.
    (Jasmine Foo/Ravi Srinivasan)
  • Strong existence and uniqueness for solutions of SDEs--case of Lipschitz coefficients.
    (Kevin Leder)
  • Examples.
    (Michael Lamar)
  • Markov properties of solutions.
    (Matt Feiszli)
  • Girsanov's theorem.
    (Tom Dean)
  • Weak existence and uniqueness via measure transformation.
    (Sergey Kushnarev)
  • Connections with partial differential equations.
    (Yingda Cheng)
  • Stochastic control.
    (Sergey Kushnarev)
  • Connections with mathematical finance.
    (Matt Feiszli)
  • Numerical solutions to SDEs. [pdf]
    (Luis Carvalho/Akil Narayan)
Additional possible topics:
  • Stochastic stability.
  • Reflected Brownian motion and reflected diffusions.
  • Numerical approximations to sample paths and expected values.
  • Stochastic integral with respect to measurable integrands and stochastic control.
  • Filtering theory.
  • Large deviations of diffusion processes.


Updated: 12/15/2005