# Course Details for A.Y. 2015/2016

#### Name:

**Probabilità e Processi Stocastici 2 / Probability and Stochastic Processes 2**

### Basic information

##### Credits:

*:* Master Degree in Mathematics 6 CFU (b)

##### Degree(s):

Master Degree in Mathematics 2^{nd} anno curriculum Generale Compulsory

##### Language:

English

### Course Objectives

The course aims to give an introduction to the theory of stochastic processes in continuous time 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 (such as Markov pure jump processes, diffusion processes) and to acquire both the mathematical tools and intuition for being able to describe systems randomly evolving in time in terms of a probability models and to analyze their properties.

### Course Content

- Continuous time stochastic processes: definition, finite dimensional distributions, stationarity, sample path spaces, construction, examples.
- Pure jump Markov processes: definition, main properties and examples.
- Poisson process with applications on queueing models. Birth and death processes. Branching processes.
- Brownian motion: definition, construction and main properties.
- Ito integral and stochastic differential equations.

### Learning Outcomes (Dublin Descriptors)

On successful completion of this course, the student should

- have knowledge of language, basic concepts and techniques of the Theory of stochastic processes, have knowledge and understanding of some relevant classes of processes (Markov processes in continuous time, Brownian Motion, Diffusions) and their properties.

have knowledge and understanding of the main mathematical tools and results on Stochastic calculus and be aware of its potential applications

evaluate the possible approaches for modeling a system with randomness using a stochastic process and be able to select the most appropriate one, to discuss its fundamental futures and to compare it with other models

demonstrate ability to describe complex systems and problems in a probabilistic framework, to explain them in terms of stochastic dynamics, to illustrate and give rigorous proofs of their main features

demonstrate capacity for reading and understanding texts and research papers on related topics

### Prerequisites and Learning Activities

Probability theory (probability spaces, conditional probability, independence, random variables and their distributions, expectation, limit theorems for sequences of random variables, martingales), real analysis, basics on measure theory and Lebesgue integral, basics on discrete times Markov chains.

### Assessment Methods and Criteria

written test

### Textbooks

- P. G. Hoel, S. C. Port, C.J. Stone, Introduction to stochastic processes , Waveland Press. 1972.
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- G. Grimmett, D. Stirzaker, Probability and Random Processes , Oxford University Press. . 2001.
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### Course page updates

This course page is available (with possible updates) also for the following academic years:

** To read the *** current * information on this course, if it is still available, go to the university course catalogue .
*Course information last updated on: 24 novembre 2015, 15:52*