Probabilità

Probability §

Preliminary material §

1. Probability spaces
2. Lemmi Borel-Cantelli
3. Lemma di Fatou
4. Random variables
5. Indipendence
6. Integration
7. Lp spaces

Conditional expectation §

1. Conditional exp on an event
2. Conditional exp on a discrete random variable
3. Conditional exp on a sigma algebra
4. Properties of conditional expectation

Discrite time martingales §

1. Introduction and motivation
2. Definition of martingale
3. Stopping times
4. The optional stopping theorem
5. Martingale convergence
6. Uniformly integrable martingales

continuous time martingales §

1. Stochastic processes in continuous time
2. Filtration and stopping times
3. continuous time martingales

Brownian motion §

1. Weak convergence of probability mesures on metric spaces
2. The Wiener mesure
3. Properties of Brownian motion