Advanced probability
- Code: DT0761
- Unit Coordinator: Ida Germana Minelli
- Programme: RealMaths
- ECTS Credits: 9
- Semester: 2
- Year: 1
- Campus: University of L'Aquila
- Language: English
- Delivery: In-class
- Aims:
The course aims to give an introduction to the theory of stochastic processes with special emphasis on applications and examples. On successful completion of this module the students should become familiar with some of the most known stochastic processes and to acquire both the mathematical tools and intuition for being able to describe systems randomly evolving in time and to analyze their properties.
- Content:
A measure theoretic approach to probability. Conditional expectation, properties, interpretations and computations. Filtrations, Martingales, examples and applications. Stopping times. The optional sampling Theorem. Martingale inequalities. Martingale convergence theorems. Uniformly integrable martingales and convergence in L^1. Continuous time processes, definition, finite dimensional distributions. Poisson process and its properties. Additive processes. The strong Markov property of Poisson process. Brownian motion: definition and main properties.
- Pre-requisites:
Basic notions of probability theory, measure theory and integration. Markov Chains.
- Reading list:
D. Williams “Probability with martingales”,
P. Billingsley “Probability and measure”,
Z. Brzezniak, T. Zastawniak “Basic stochastic processes”
