This course starts with a review of probability theory (random variables, distributions, characteristic functions, limit theorems and notions of convergence, etc.). Then it introduces the definition and classifications of stochastic processes, and the basic concepts of stationary independent increments, martingales, Markov processes, stationary processes, renewal processes, point processes, et cetera. After that it covers in depth the discrete-time Markov chains (transition probability matrices, the Chapman-Kolmogorov equations, classification of states, recurrence, limit theorems, random walks, etc.), continuous-time Markov chains (Poisson processes, birth death processes, renewal processes, etc.), and martingales (supermartingales and submartingales, optional sampling/stopping theorem, etc.). This course will also cover applications of the random processes in areas such as queuing theory or stock markets. Some advanced topics may be included.

Advisory note: Students are expected to have a basic background in probability theory.