The course will follow selected topics from the book of Le Gall
https://www.amazon.com/Brownian-Martingales-Stochastic-Calculus-Mathematics/dp/3319310887

Among the topics discussed: Gaussian processes, white noise, Brownian motion and its basic properties, continuous-time martingales and their properties, stochastic integrals (Itô integral) with respect to continuous martingales and semimartingales.

Time permitting, we will include short discussions of further properties of Brownian motion such as recurrence and transience, conformal invariance, the relation of Brownian motion and partial diffferential equations, and stochastic differential equations.

Brownian motion and stochastic calculus form classical topics in probability theory which have found many applications in mathematics, physics, engineering, economocis and other areas. Due to the time constraints, applications will not be discussed beyond the topics mentioned above but the theoretical background gained will allow the interested student to pursue further applications.

Prerequisites: Probability for Mathematicians and Complex Functions.

The course is suitable also for advanced undergraduate students who have studied the prerequisites.