The course is organised as a series of classes where the students will have the possibility to solve and discuss the solutions of a series of simple and more advanced exercises on the theory of stochastic processes.  

The students will acquire autonomy in studying more advanced theoretical aspects of stochastic processes and in applying the main ideas of stochastic processes to the subsequent studies in the area of statistics and finance.  

Program

Random walks. Exercises on: Markovianity, temporal and spatial invariance, random walks with reflecting and absorbing barriers, reflection principle, ballot theorem, distribution of the maximum, hitting time theorem, first and second arc sine law, random walks and generating functions.    

Brownian motion. Exercises on: path properties of Brownian motion, Brownian motion as a strong Markov process, transience and recurrence. 

Branching processes. Exercises on: expectation and variance of the population size, probability of extinction of the population.   

Markov chains. Exercises on: transition matrix, classification of states, classification of chains, stationary distribution and limit theorem, chains with finitely many states.  

Poisson processes. Exercises on the main properties of Poisson processes.  

Stationary processes. Exercises on: variance and covariance function, linear predictions, spectral theorem for autocorrelation functions, ergodic theorem.   

Recommended books:

- G.R. Grimmett and D.R. Stirzaker. One Thousand Exercises in Probability. 3rd edn, OUP, 2001

- P. Mörters and Y. Peres. Brownian Motion. Cambridge Series in Statistical and Probabilistic Mathematics, 2010