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Stochastic calculus
MA 598
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Introduction
The central object of this course is Brownian motion. This stochastic process (denoted by W in the sequel) is used in numerous concrete situations, ranging from engineering to finance or biology. It is also of crucial interest in probability theory, owing to the fact that this process is Gaussian, martingale and Markov at the same time. This very rich structure converts Brownian motion into a fascinating object, but it should also be pointed out that the paths of W are irregular (in particular nowhere differentiable). Our first aim will thus be a good description of Brownian motion. We will then construct a differential calculus with respect to W. Specifically, our main tasks will be:
  1. Give a formula for the differential df(W) when f is smooth enough. This fundamental relation is called Itô's formula.
  2. Solve differential equations driven by W, and give elementary properties for their solutions.
  3. If time constraints allow it, give an introduction to the analysis on Wiener's space.
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Bibliography
  1. R. Durrett: Stochastic calculus. A practical introduction. Probability and Stochastics Series. CRC Press, 1996.
  2. I. Karatzas, S. Shreve: Brownian motion and stochastic calculus. Second edition. Graduate Texts in Mathematics, 113. Springer-Verlag, 1991.
  3. D. Revuz, M. Yor: Continuous martingales and Brownian motion. Third edition. Springer-Verlag, 1999.
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Homework
We will follow a problem list. Homeworks are usually due on tuesday after class, every two weeks.
The assignment is below.
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Problems Due
3-6-7-8-9 1/26
16-22-23-25-31 2/16
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There will also be some reading projects specified in the link.
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Office hours
2:00-3:30 on Tuesday, in Math 434.
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Slides
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