Basics of Stochastic Processes
Math 595, CRN 45995, Section BSP
This is a short course (January 16th to March 7th) which quickly introduces the basics
of the modern theory of stochastic processes. We shall emphasize
those parts of the theory which are useful in a wide variety
of disciplines. Our intended audience is not only mathematicians,
but students from engineering, physics, and finance. We assume that
the students will either be willing to accept, ex cathedra, basic aspects
of measure theory, or have the ability to understand them on their own.
Provisional Schedule
- Lecture 1: Review of measure theory and random variables
- Lecture 2: Brownian motion
- Construction
- Regularity
- Scaling
- Martingale and Markov properties
- Lecture 3: Ito theory
- Ito integrals
- Ito's formula
- Levy's characterization of Brownian motion
- Feynman-Kac formulae
- Girsanov's formula
- Risk-Neutral Probabilities
- Burkholder inequalities
- Lecture 4: Stochastic differential equations
- Construction, Existence and Uniqueness
- Markov property
- 1-dimensional SDE's (Feller's test for explosions)
- generator
- Fokker-Planck equation
- Lecture 5: Martingale formulation of Markov processes
Grading: Grades will
be determined on the basis of homework (30%), a midterm (30%) and a
final (40%).
Text: Oksendal, Stochastic Differential Equations: An Introduction with Applications, 2003, Springer-Verlag
Additional References: Karatzas and Shreve, Brownian Motion and Stochastic Calculus, 2nd
ed., 2005, Springer-Verlag.
Instructor: Richard Sowers
Office: 347 Illini Hall
Phone: (217) 333-6246
email: r-sowers@math.uiuc.edu
Home page: http://www.math.uiuc.edu/~r-sowers
Class Time: Mondays, Wednesdays, and Fridays 12-12:50 P.M., 148 Henry