Stochastic processes

About the Course

The course relies on the basic knowledge in the following disciplines: probability theory, calculus and linear algebra.

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

Reading materials for the course contain 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, some of which contain non-graded questions. Each week is followed by a test with both theoretical and practical problems related to the covered material. At the end of the course students are encouraged to complete the final exam, which comprises various problems on all the topics discussed during the lectures

Course Objectives

Introduce the main concepts of the theory of stochastic processes

Provide ideas on the application of stochastic processes to the solution of various problems in economics, finance, and other related fields

Provide solid theoretical basis for studying further disciplines in stochastics, such as stochastic modelling and financial mathematics

Learning Outcomes

1. Get acquainted with the main types of stochastic processes, with the basic notions in stochastic processes, such as stationarity and ergodicity

2. Be able to study the most important properties of stochastic processes

3. Be introduced to the possible applications of the considered models

4. Be able to choose an appropriate process for describing real-life problems in various quantitative fields

Course Syllabus

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.

Week 9. Final exam

This module includes final exam covering all topics of this course.