Homeworks: there will be four homework sets, as well as a take-home final exam.
The first task will be to understand conditional expectation. This is harder than you think. From there, we will define and study martingales. We will discuss Markov chains, not giving a comprehensive treatment but hitting the basics and some highlights. Lastly we will construct Brownian motion and just barely scratch the surface as to the properties of this strange object.
"Probability: theory and examples", 3rd Edition, by R. Durrett.