# Stochastic Calculus (Fall 2012)

From Lecture Notes

**Please help edit and improve these notes.**

Anyone can edit this wiki, but you must log in/create an account first (top right). Here are some editing guidelines:

- Create a sub-page for each lecture, following the current naming convention: E.g.
`{{FULLPAGENAME}}/Lecture 1`

. - Include the Template:Stochastic Calculus pages into every lecture page using the line
{{Stochastic Calculus}}

- Break up topics in each lecture using sections.
- Start / update / edit a Collection arranging these articles into a logical order. (This will allow for PDF printing of the whole course, if I can make it work; Currently you can only print each lecture individually.)

This is based on a Stochastic Calculus course taught in Fall 2012 by Gautam Iyer. Ryan Murray adapted his real time LaTeX notes into this Wiki.

- Course Outline and References.
- Lecture 1: Stochastic Processes, Filtrations and Stopping Times.
- Lecture 2: Martingales in Continuous Time.
- Lecture 3: Optional Sampling Theorem.
- Lecture 4: Construction of Quadratic Variation
- Lecture 5: Further development of Quadratic Variation
- Lecture 6: Construction of Brownian Motion
- Lecture 7: A Better Construction of Brownian Motion
- Lecture 8: The Ito Integrals: Ito Isometry.
- Lecture 9: Approximation by Simple Processes, quadratic variation, and Martingale Characterization.
- Lecture 10: Ito's Formula, and Levy's criterion.
- Lecture 11: The Proof of Ito's Formula, and an application to exit times.
- Lecture 12: Martingale and Ito representation theorems
- Lecture 13: Martingale and Ito representation theorems (continued)
- Lecture 14: Markov and Strong Markov property for Brownian Motion.
- Lecture 15: Reflection principle, passage time densities of Brownian motion.
- Lecture 16: Running maximum of Brownian Motion, and the law of iterated logarithm.
- Lecture 17: Girsanov Theorem
- Lecture 18: Passage times of Brownian Motion with drift, and regularity of exponential martingales.
- Lecture 19: SDE's: Strong solutions
- Lecture 20: SDE's: Weak solutions
- Lecture 21: Diffusions: Strong Markov Property.
- Lecture 22: Dynkin's formula, Generators, and Recurrence of Brownian Motion.
- Lecture 23: Kolmogorov Equations, Feynman Kac formula.
- Lecture 24: Dirichlet-Poisson problem in Bounded Domains.

## Transcription Resources

- File:880-2012-notes.tex: Ryan Murray's real time transcription of lectures.