250157 VO Stochastic Analysis (2021W)

This course aims at rigorously developing Ito's theory of stochastic calculus and presenting some of its fundamental applications.

Some of the keywords are: Gaussian processes, Brownian motion, conditional expectation, martingales, stopping times, optional stopping, local martingales, stochastic integral, Ito's lemma.

We will first construct Brownian motion and derive its basic properties. Then, we will develop a formal theory of continuous martingales and local martingales, on which we will build the stochastic integral.

Towards the end of the course we will use the constructed theory of stochastic calculus to derive some deep results on the nature of Brownian motion (like for example conformal invariance of two-dimensional Brownian motion).

Familiarity with Advanced Probability will be assumed. However, we will recall the notion of conditional expectation. Elements of complex analysis will be used towards the end of the course.