The lecture covers the following topics:

Stochastic Processes
Examples of stochastic processes, filtrations and conditional expectations, stopping times and hitting times.

Brownian Motion
Brownian motion and Gaussian processes, path properties of Brownian motion, the Kolmogorov consistency theorem, the Kolmogorov-Centsov continuity theorem, existence of Brownian motion.

Martingales
Definition and examples, discrete time martingales, paths of continuous martingales.

Martingale Convergence
Upcrossings and submartingale limits, uniform integrability and Vitali’s theorem, the martingale convergence theorem, continuous time martingales.

Markov Chains
Markov property and transition probabilities, the strong Markov property, classification of states, invariant measures and stationary distributions, convergence to the stationary distribution.