# Course Details

#### Name:

**Stochastic Processes / Stochastic Processes**

### Basic information

##### Credits:

*Master Degree in Mathematical Engineering:* 6 Ects (b)

*Master Degree in Mathematics:* 6 Ects (c)

##### Degree(s):

Elective 2^{nd} year Master Degree in Mathematics curriculum Generale

Elective 1^{st} year Master Degree in Mathematical Engineering curriculum Comune

##### Language:

English

### Course Objectives

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 classes of stochastic processes (such as martingales, markov processes, diffusion processes) and to acquire both the mathematical tools and intuition for being able to describe systems with randomness evolving in time in terms of a probability model and to analyze it charcterizing some of its properties

### Course Content

- Stochastic Processes: definition, finite dimensional distributions, stationarity, sample path spaces, construction, examples
- Filtrations, stopping times, conditional expectation.
- Markov processes: definition, main properties and examples. Birth and death processes.
- Poisson process with applications on queueing models.
- Martingales: definition, main properties and examples.
- Brownian motion: definition, construction and main properties.
- Brownian Bridge, Geometric Brownian Motion, Ornstein-Uhlenbeck process.
- Ito integral and stochastic differential equations. Applications and examples

### 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, Martingales, Diffusions) and their properties, have knowledge and understanding of tha main mathematical tools and results on Stochastic calculus and be aware of its potential applications
- be able to identify, analyse and prove relevant properties of models based on stochastic processes, be able to solve problems related to such models
- evaluate the possible approaches to modeling a system with randomness using a stochastic process, 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 way, 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, product spaces, random variables and their distributions, expectation, convergence, limit theorems for sequences of random variables), real analysis, basics on measure theory and Lebesgue integral, basics on discrete times markov chains

### Teaching Methods

**Language**: English

Lectures and exercises

### Assessment Methods and Criteria

Written and oral exam.

### Textbooks

- J. Rosenthal,
**A first look at Rigorous Probability Theory**. World Scientific. *Additional text *
- G. Grimmett, D. Stirzaker,
**Probability and random Processes**. Oxford University Press. * *
- B. Oksendal,
**Stochastic Differential Equations**. Springer-Verlag . * *
- G. Lawler,
**Introduction to Stochastic Processes**. Chapman & Hall. * *
- Z. Brzezniak and T. Zastawniak,
**Basic Stochastic Processes, A Course Through Exercises**. Springer. * *

### Online Teaching Resources

### Recent teaching material

This list contains only the latest published resources. Resources marked with an asterisk belong to other courses (indicated between brackets)

### Course page updates

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

*Course information last updated on: 24 febbraio 2018, 01:39*