Statistical Methods and Applications

VALENTINA CAMMAROTA

Learning goals
The course provides a broad introduction to stochastic processes. In particular the aim is
- to give a rigorous introduction to the theory of stochastic processes,
- to discuss the most important stochastic processes in some depth with examples and applications,
- to give the flavour of more advanced work and applications,
- to apply these ideas to answer basic questions in several applied situations including biology, finance and search engine algorithms.

Knowledge and understanding
At the end of the course the students will be familiar with the basic concepts of the theory of stochastic processes in discrete and continuous time and will be able to apply various techniques to study stochastic models that appear in applications.

Applying knowledge and understanding
At the end of the course the students will have the tools to grasp and formalize, in the language of stochastic processes, phenomena that evolve in time and space. The students will have the tools 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. The student will also acquire the necessary language skills to start reading academic books on the topic and research papers.

Communication skills
The students will acquire the intuition and the communication skills necessary to describe phenomena in the mathematical language of stochastic processes. In particular the student will also acquire the rationale behind the stochastic model studied (e.g. the ideas of Markovianity, transience, recurrence, equilibrium, stationarity, long and short-time behaviour...) that is necessary to communicate to specialist and non-specialist audiences.

Learning skills
The students will acquire the methodology and the language to study in a manner that may be largely autonomous and to apply the methodology to the subsequent studies in the area of statistics and finance.

Learning goals
The course provides a broad introduction to stochastic processes. In particular the aim is
- to give a rigorous introduction to the theory of stochastic processes,
- to discuss the most important stochastic processes in some depth with examples and applications,
- to give the flavour of more advanced work and applications,
- to apply these ideas to answer basic questions in several applied situations including biology, finance and search engine algorithms.

Prerequisites:

- probability theory,
- elementary analysis (measure theory and linear functional analysis),
- ordinary and partial differential equations,
- linear algebra.

Random walks (about 18 hours)
Definition, 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, short introduction on Black–Scholes model.


Brownian motion (about 18 hours)
Definition and existence, Brownian motion as a limit of a simple random walk, path properties of Brownian motion, Brownian motion as a strong Markov process, transience and recurrence.


Branching processes (about 6 hours)
Definition, expectation and variance of the population size, geometric branching, probability of extinction of the population.


Markov chains (about 18 hours)
Definition, homogeneous Markov chains, transition matrix, examples of Markov chains, classification of states, classification of chains, stationary distribution and limit theorem, chains with finitely many states, short introduction on Monte Carlo method,
MCMC and search engine algorithms.


Poisson processes (about 6 hours)
Definition and main properties.

Recommended books:

- G.R. Grimmett and D.R. Stirzaker. Probability and Random Processes. 3rd edn, OUP, 2001

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


Helpful books:

- D. Williams. Probability with Martingales. CUP, 1991


Teaching material is also delivered through e-learning platform "moodle" at the following address

https://elearning2.uniroma1.it/course/view.php?id=6395

- G.R. Grimmett and D.R. Stirzaker. Probability and Random Processes. 3rd edn, OUP, 2001

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

- D. Williams. Probability with Martingales. CUP, 1991

The course is organised in a series of taught classes that will present the theoretical aspects, through rigorous proofs, of the main result in the area with examples and applications.

The attendance at the course is strongly recommended. The students will find, during the course, the updated program with book references for each topic on E-learning. The students that will not be able to attend the course are strongly recommended to contact the professor.

The examination consists of three questions on three different topics studied during the course. The students are required to answer the questions during the written exam showing that they have acquired the intuition and the technical language, and that they are able to present the topics and the proofs with all necessary details. The oral exam consists in a discussion of the written exam. 80% of the final grade is based on the written exam.

1) Random walks
• Derive E[Nb] where Nb is the number of visits to b before the random walk returns to its
starting point.
• Explain the meaning of the result E[Nb] = 1.
• State and prove the Arcsin law for the last visit to the origin.
2) Brownian motion
• State and prove the strong Markov property for Brownian motion.
• State and prove the reflection principle for Brownian motion.
• Derive the distribution of the maximum of Brownian motion.
3) Markov chains
• Let ρi(x) the mean number of visits of the chain to the state i between two successive visits
to the state k. Prove that for any state k of an irreducible persistent chain the vector
ρ(k) = (ρi(x), i ∈ S) satisfy ρ(k) = ρ(k) · P.
• State and discuss the theorem that gives the link between the existence of a stationary distribution and the limiting behaviour of the probabilities pi,j (n) ad n