Introduction

This course introduces various crucial notions concerning discrete and continuous time random functions. It has to be seen as a continuation of MA 538.

We begin by a review on Gaussian vectors and the central limit theorem (a leftover from the Spring 18 MA 538 course). Next we study the fundamental and delicate concept of conditional expectation. We will then introduce the notion of martingale, which is a class of stochastic processes arising in the description of fair games. We will study the convergence properties of this kind of object, when the time index is discrete. The next topic to be covered (if our schedule allows it) concerns ergodic theorems, which can be seen as another tool allowing to get limit theorems for sequences of random variables. Eventually we construct and analyze the most important continuous time stochastic process, namely Brownian motion. According to time, we will also cover some basic notions of stochastic calculus such as Itô's formula.

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Bibliography

1. R. Durrett: Probability: theory and examples. Second edition. Duxbury Press, 1996.
2. S. Resnick: A probability path. Birkhaüser-Springer, 2014.
3. D. Williams: Probability with martingales. Cambridge University Press, 1991.
4. R. Durrett: Stochastic calculus. A practical introduction. Probability and Stochastics Series. CRC Press, 1996.
5. I. Karatzas, S. Shreve: Brownian motion and stochastic calculus. Second edition. Graduate Texts in Mathematics, 113. Springer-Verlag, 1991.

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Homework

We will follow a problem list. Homeworks are usually due on tuesday after class.
The assignment is given in the attached calendar.

• Ground rules
• Gaussian vectors and CLT
• Conditional expectation
• Martingales
• Brownian motion
• Itô's formula
• SDEs
• Ergodic theorems