# 30515 - APPLIED STOCHASTIC PROCESSES

Course taught in English

Go to class group/s: 31

Solid knowledge of calculus. Good understanding of basic probability theory (e.g. random variables) and basic programming tools.

Stochastic processes are the natural tool to model real-world phenomena involving randomness and uncertainty. They offer a power mathematical framework to analyze complex problems in a variety of applied areas, ranging from business and industry to economics, finance, social sciences, biology and computer science. Moreover, the use of stochastic processes (and related probabilistic techniques) to build advanced statistical models is central to the ongoing data science revolution. The aim of the course it to provide students with a basic understanding of the probabilistic models and techniques underlying the most widely used classes of stochastic processes. The main focus is on modeling aspects, which are completed by a description of some popular algorithms for simulation. Mathematical concepts are integrated with real-world applications and examples. At the end of the course, students have bridged the gap between their elementary probability skills and the knowledge required to understand and use basic models based on stochastic processes.

- Conditional probabilities and conditional expectations.
- Markov chains: Chapman-Kolmogorov equation, Classification of states, Limiting properties, Applications (e.g. website ranking).
- Markov Chain Monte Carlo methods. Computational applications (e.g. big data analysis).
- Counting processes and the Poisson process. Examples and modeling applications.
- Simulation techniques. Rejection sampling, importance sampling.

- Understand the connection between simple random variables and more elaborate and realistic stochastic processes.
- Formulate and analyse probabilistic models based on Markov chains and counting processes.
- Understand the principles of stochastic simulation and Monte Carlo algorithms.

- Translate real-world situations involving randomness and uncertainty into probabilistic models.
- Exploit the tools of probability theory to answer relevant questions, such as: what is the equilibrium distribution of a specific stochastic system? What is the probability of a certain outcome?
- Use software to simulate the evolution of probabilistic systems. Design simple Monte Carlo algorithms for computational purposes.
- Apply stochastic processes to model and solve real-world problems commonly occurring in diverse fields, such as business, industry, economics, finance and data science.

- Face-to-face lectures
- Exercises (exercises, database, software etc.)

The main learning activity of this course consists in face-to-face lectures. Lectures integrate discussions of theoretical notions, such as mathematical definitions, with examples and in-class solution of exercises. The aim of the examples and exercises shown in class is two folded:

- First to illustrate and visualize the theoretical notions with simple examples.
- Secondly to describe real-world applications (both modeling and computational) of the probabilistic tools.

In addition, the R statistical software is used in class to display computer simulations of the stochastic models under consideration. A set of exercises are provided for students to practice and get familiar with the lectures’ content.

Continuous assessment | Partial exams | General exam | |
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The assessment for all students, both attending and non-attending, consists in a written exam. The aim of the exam is to verify both the students’ understanding of the theoretical material (e.g. ability to provide definitions and characterizations of some noteworthy stochastic process), and their capacity to apply the theory for modeling or computational purposes (e.g. ability to identify appropriate models for a given situation and use them to answer specific questions).

S.M. ROSS, *Introduction to probability models,* Academic Press, 2014.