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Department of Mathematics and Applications
"Renato Caccioppoli"


Stochastic Processes

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Stochastic Processes

Credits

6

Prerequisites

None.

Scientific-disciplinary sector (SSD)

MAT/06 Probability and Mathematical Statistics.

Examination method

Oral exam.

Learning
objectives

The course aims to strengthen basic knowledge of Probability Theory (while at the same time making the class more homogeneous) by revisiting fundamental contents in a markedly formal way. It provides concepts, contents and tools, such as definitions, properties and theorems concerning conditional expectations, stopping times, martingales, Brownian motion and stochastic integration, which form the basis both for a more in-depth study of the theory and for an informed use of stochastic processes in applications.

Syllabus

Review of fundamental definitions and theorems of probability measure theory. Conditional expectations, with numerous examples of applications. Stopping times. Martingales and convergence results. Examples. Brownian motion and Brownian bridge. Notable laws of Brownian motion. Analytical approach to Brownian motion. Stochastic integration. Itô’s formula and stochastic differential equations.

Expected learning
outcomes

By the end of the course, students must demonstrate that they

  • know and understand the theoretical foundations of the stochastic processes covered in the lectures, with particular regard to stopping times, martingales, Brownian motion and stochastic integration;
  • are able to apply the knowledge acquired to solve exercises and problems of varying complexity independently;
  • are able to communicate ideas and solutions clearly, rigorously and effectively to both specialist and non-specialist audiences;
  • are able to identify the most appropriate methods to analyse and solve a problem relating to the course topics and to interpret the results correctly.

Learning outcomes
to be assessed

The criteria for assessing learning and for grading are, in order: clarity, correctness and completeness of exposition; ability to develop simulation algorithms.