Stochastic processes

This course is aimed at the students with any quantitative background, such as
— Pure and applied mathematics
— Engineering
— Economics
— Finance
and other related fields.

The present course introduces the main concepts of the theory of stochastic processes and its applications. During the study, the students will get acquainted with various types of stochastic processes and learn to analyse their basic properties and characteristics. The material is anticipated to be of great interest for students willing to enhance their knowledge of stochastics and its use for the analysis of complex dynamical systems arising in various fields, such as economics or engineering.

The main purpose of this course is to introduce the main concepts of the theory of stochastic processes and provide some ideas for its application to the solution of various problems in economics, finance, and other related fields.

The course relies on the basic knowledge in the following disciplines:
— probability theory (e.g., discrete and continuous distributions, conditional probability, calculation of moments, covariance, basic characteristics of functions of random variables)
— calculus (e.g., integration, double integration, differentiation, trigonometry)
— linear algebra (solution of systems of linear equations)
Acquaintance with the basics of mathematical statistics is not required but simplifies the understanding of this course.

Each week is followed by a test containing both theoretical and practical problems related to the covered material. At the end of the course the students are encouraged to complete the final exam, which comprises various problems on all the topics discussed during the lectures.

No specific software is needed for the completion of this course.

The course provides a solid theoretical basis for studying further disciplines in stochastics, such as stochastic modelling and financial mathematics. In addition, the reading materials contain the examples of real-life applications of the studied concepts, which might be helpful for designing the own solutions for various problems arising in scientific research, business and other areas.

The course consists of short video lectures, up to 20 minutes long, some of which contain non-graded questions which enhance the understanding of the material. Each week there is a test with an estimated completion time of 1 hour. The final exam consists of test problems covering all the material and is expected to take approximately 1.5 hours to complete.

• Week 1: Introduction & Renewal processes
• Upon completing this week, the learner will be able to understand the basic notions of probability theory, give a definition of a stochastic process; plot a trajectory and find finite-dimensional distributions for simple stochastic processes. Moreover, the learner will be able to apply Renewal Theory to marketing, both calculate the mathematical expectation of a countable process for any renewal process
• Week 2: Poisson Processes
• Upon completing this week, the learner will be able to understand the definitions and main properties of Poisson processes of different types and apply these processes to various real-life tasks, for instance, to model customer activity in marketing and to model aggregated claim sizes in insurance; understand a relation of this kind of models to Queueing Theory
• Week 3: Markov Chains
• Upon completing this week, the learner will be able to identify whether the process is a Markov chain and characterize it; classify the states of a Markov chain and apply ergodic theorem for finding limiting distributions on states
• Week 4: Gaussian Processes
• Upon completing this week, the learner will be able to understand the notions of Gaussian vector, Gaussian process and Brownian motion (Wiener process); define a Gaussian process by its mean and covariance function and apply the theoretical properties of Brownian motion for solving various tasks
• Week 5: Stationarity and Linear filters
• Upon completing this week, the learner will be able to determine whether a given stochastic process is stationary and ergodic; determine whether a given stochastic process has a continuous modification; calculate the spectral density of a given wide-sense stationary process and apply spectral functions to the analysis of linear filters.
• Week 6: Ergodicity, differentiability, continuity
• Upon completing this week, the learner will be able to determine whether a given stochastic process is differentiable and apply the term of continuity and ergodicity to stochastic processes
• Week 7: Stochastic integration & Itô formula
• Upon completing this week, the learner will be able to calculate stochastic integrals of various types and apply Itô’s formula for calculation of stochastic integrals as well as for construction of various stochastic models.
• Week 8: Lévy processes
• Upon completing this week, the learner will be able to understand the main properties of Lévy processes; construct a Lévy process from an infinitely-divisible distribution; characterize the activity of jumps of a given Lévy process; apply the Lévy-Khintchine representation for a particular Lévy process and understand the time change techniques, stochastic volatility approach are other ideas for construction of Lévy-based models.
• Final exam
• This module includes final exam covering all topics of this course