# Stochastic Calculus (Fall 2012)

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

1. Create a sub-page for each lecture, following the current naming convention: E.g. `{{FULLPAGENAME}}/Lecture 1`.
2. Include the Template:Stochastic Calculus pages into every lecture page using the line
`{{Stochastic Calculus}}`
3. Break up topics in each lecture using sections.
4. 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.