Math 606, Summer 2021.
Topics in Stochastic Processes (054–39482R)
Singular Stochastic Ordinary Differential Equations
Class meetings: MW, 9:30am-12:30pm, On line.
Information on this and related pages changes frequently.
Instructor: Sergey Lototsky
Office: KAP 248D.
Phone: 213 740 2389
E-mail: lototsky usc edu.
URL: https://dornsife.usc.edu/sergey-lototsky/
Office Hours: MW before and after the class. Appointments at other time are welcome.
Course objective To understand one-dimensional SODEs beyond the standard existence/uniqueness theorem
Course work: Class participation, homework assignments, final presentation.
Official grading scheme: 20% class participation, 40% homework assignments, 40% final presentation.
Main reference: Alexander S. Cherny and Hans-Jürgen Engelbert. Singular stochastic differential equations. Lecture Notes in Mathematics, 1858. Springer-Verlag, Berlin, 2005. viii+128 pp. ISBN: 3-540-24007-1. The book is available in electronic form through the the USC Libraries, for example, by following the MathSciNet link.
The main file containing the homework problems and some ideas to remember; the file will be constantly updated as we proceed with the class.
Additional references:
- O. Kallenberg, Foundations of Modern Probability (Springer, 1997), Chapters 19 amd 20.
- S. Karlin and H. M. Taylor, A Second Course in Stochastic Processes (Academic Press, 1981), Chapter 15.
- I. Karatzas and S. Shreve, Brownian Motion and Stochastic Calculus, Second Edition (Springer, 1991), Sections 3.6, 3.7, and 5.5.
My notes:
- Stochastic analysis in continuous time
- A summary of Brownian motion
- A summary of SODEs
- A summary of local time
- A summary of Markov processes
- A summary of ergodic theory
- A summary of the Girsanov theorem and related results
- A summary of diffusions in one dimension
Other notes:
- Delbaen and Shirakawa on dX=B(X)dW
- A strict local martingale
Our progress:
May 19: General introduction and examples of singular SODEs (interest rate models, Bessel processes, continuum versions of the Wright-Fisher model in biology)
May 24: Survey of stochastic analysis in continuous time; strong and weak solutions of SODEs.
May 26: What combinations of weak/in distribution and strong/pathwise are possible? Scale function and removal of drift. OU, GBM, and other affine equations. Bernoulli SODE and an example of local existence on a random time interval.
May 31: Memorial Day; no class.
June 2: Constructing a non-zero solution of dX=X^{1/3}dW, X(0)=0
June 7: A close look at dX=B(X(t))dW(t), B^2(x)>0, x>0.
June 9: The equation dX=B(X(t))dW(t) in general: the theorems of Engelbert and Schmidt.
June 14: One-dimensional diffusions: characterizations and the canonical form of the generator
June 16: Boundary points, boundary conditions, and Feller’s criteria for (non)explosion
June 21: Ergodicity and Khasminskii’s criteria for (non)explosion.
June 23: Comparative analysis of Feller’s and Khasminskii’s criteria for (non)explosion.
June 28: KL expansion and related topics; a summary of strong solutions.
June 30: A (problem-solving) discussion.