Learning goals.

General learning targets: 

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 advanced exercises on the theory of stochastic processes.  

Knowledge and understanding.

At the end of the course the students will acquire the ability to solve autonomously simple and more advanced exercises on the theory of stochastic processes.  

Applying knowledge and understanding.

During the course the students will exercise the ability to grasp and formalize, in the language of stochastic processes, phenomena that evolve in time and space and to solve simple applied problems in new environments and broader contexts. 

Making judgements At the end of the course the students will have the tools to evaluate critically and choose between different stochastic models to model phenomena that evolve in time and space.  

Communication skills.

The students will exercise the intuition and the communication skills necessary to describe phenomena in the mathematical language of stochastic processes. 

In particular the student will acquire familiarity with the main ideas that are behind the stochastic model, e.g., the ideas of Markovianity, transience, recurrence, equilibrium, stationarity, long and short-time behaviour. Learning skills.

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.

Prerequisites:

- elementary probability theory,

- analysis (measure theory and linear functional analysis). 

- ordinary and partial differential equations,

- linear algebra.  

Random walks (about 6 hours)

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 (about 6 hours)  

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

Branching processes (about 2 hours)

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

Markov chains  (about 6 hours)

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

Poisson processes (about 2 hours) 

Exercises on the main properties of Poisson processes.  

Stationary processes (about 2 hours)

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

The course is organised in a series of taught 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 are required to prepare a short written essay where they solve 4 exercises on the topics studied during the course. 

In the solution of the exercises the students must show that they have acquired the techniques to solve the problems, the intuition, the technical language, and that they are able to present the topics with all necessary details.