Offered By

National Research University Higher School of Economics

About this Course

25,523

The purpose of this course is to equip students with theoretical knowledge and practical skills, which are necessary for the analysis of stochastic dynamical systems in economics, engineering and other fields.
More precisely, the objectives are
1. study of the basic concepts of the theory of stochastic processes;
2. introduction of the most important types of stochastic processes;
3. study of various properties and characteristics of processes;
4. study of the methods for describing and analyzing complex stochastic models.
Practical skills, acquired during the study process:
1. understanding the most important types of stochastic processes (Poisson, Markov, Gaussian, Wiener processes and others) and ability of finding the most appropriate process for modelling in particular situations arising in economics, engineering and other fields;
2. understanding the notions of ergodicity, stationarity, stochastic integration; application of these terms in context of financial mathematics;
It is assumed that the students are familiar with the basics of probability theory. Knowledge of the basics of mathematical statistics is not required, but it simplifies the understanding of this course.
The course provides a necessary theoretical basis for studying other courses in stochastics, such as financial mathematics, quantitative finance, stochastic modeling and the theory of jump - type processes.

Start instantly and learn at your own schedule.

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Suggested: 8 weeks of study, 6-8 hours per week...

Subtitles: English

Start instantly and learn at your own schedule.

Reset deadlines in accordance to your schedule.

Suggested: 8 weeks of study, 6-8 hours per week...

Subtitles: English

Week

1Upon 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...

12 videos (Total 88 min), 1 quiz

Welcome1m

Week 1.1: Difference between deterministic and stochastic world4m

Week 1.2: Difference between various fields of stochastics6m

Week 1.3: Probability space8m

Week 1.4: Definition of a stochastic function. Types of stochastic functions.4m

Week 1.5: Trajectories and finite-dimensional distributions5m

Week 1.6: Renewal process. Counting process7m

Week 1.7: Convolution11m

Week 1.8: Laplace transform. Calculation of an expectation of a counting process-17m

Week 1.9: Laplace transform. Calculation of an expectation of a counting process-26m

Week 1.10: Laplace transform. Calculation of an expectation of a counting process-38m

Week 1.11: Limit theorems for renewal processes14m

Introduction & Renewal processes12m

Week

2Upon 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...

17 videos (Total 89 min), 1 quiz

Week 2.2: Definition of a Poisson process as a special example of renewal process. Exact forms of the distributions of the renewal process and the counting process-23m

Week 2.3: Definition of a Poisson process as a special example of renewal process. Exact forms of the distributions of the renewal process and the counting process-34m

Week 2.4: Definition of a Poisson process as a special example of renewal process. Exact forms of the distributions of the renewal process and the counting process-44m

Week 2.5: Memoryless property5m

Week 2.6: Other definitions of Poisson processes-13m

Week 2.7: Other definitions of Poisson processes-24m

Week 2.8: Non-homogeneous Poisson processes-14m

Week 2.9: Non-homogeneous Poisson processes-24m

Week 2.10: Relation between renewal theory and non-homogeneous Poisson processes-14m

Week 2.11: Relation between renewal theory and non-homogeneous Poisson processes-27m

Week 2.12: Relation between renewal theory and non-homogeneous Poisson processes-34m

Week 2.13: Elements of the queueing theory. M/G/k systems-19m

Week 2.14: Elements of the queueing theory. M/G/k systems-25m

Week 2.15: Compound Poisson processes-16m

Week 2.16: Compound Poisson processes-26m

Week 2.17: Compound Poisson processes-33m

Poisson processes & Queueing theory14m

Week

3Upon 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...

7 videos (Total 73 min), 1 quiz

Week 3.2: Matrix representation of a Markov chain. Transition matrix. Chapman-Kolmogorov equation11m

Week 3.3: Graphic representation. Classification of states-110m

Week 3.4: Graphic representation. Classification of states-24m

Week 3.5: Graphic representation. Classification of states-37m

Week 3.6: Ergodic chains. Ergodic theorem-16m

Week 3.7: Ergodic chains. Ergodic theorem-215m

Markov Chains12m

Week

4Upon 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...

8 videos (Total 87 min), 1 quiz

Week 4.2: Gaussian vector. Definition and main properties19m

Week 4.3: Connection between independence of normal random variables and absence of correlation13m

Week 4.4: Definition of a Gaussian process. Covariance function-15m

Week 4.5: Definition of a Gaussian process. Covariance function-210m

Week 4.6: Two definitions of a Brownian motion18m

Week 4.7: Modification of a process. Kolmogorov continuity theorem7m

Week 4.8: Main properties of Brownian motion6m

Gaussian processes12m

Week

5Upon 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....

8 videos (Total 78 min), 1 quiz

Week 5.2: Two types of stationarity-28m

Week 5.3: Spectral density of a wide-sense stationary process-17m

Week 5.4: Spectral density of a wide-sense stationary process-24m

Week 5.5: Stochastic integration of the simplest type10m

Week 5.6: Moving-average filters-15m

Week 5.7: Moving-average filters-212m

Week 5.8: Moving-average filters-38m

Stationarity and linear filters12m

Week

6Upon 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...

4 videos (Total 53 min), 1 quiz

Week 6.2: Ergodicity of wide-sense stationary processes15m

Week 6.3: Definition of a stochastic derivative11m

Week 6.4: Continuity in the mean-squared sense9m

Ergodicity, differentiability, continuity10m

Week

7Upon 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....

10 videos (Total 82 min), 1 quiz

Week 7.2: Integrals of the type ∫ f(t) dW_t-113m

Week 7.3: Integrals of the type ∫ f(t) dW_t-211m

Week 7.4: Integrals of the type ∫ X_t dW_t-15m

Week 7.5: Integrals of the type ∫ X_t dW_t-214m

Week 7.6: Integrals of the type ∫ X_t dY_t, where Y_t is an Itô process6m

Week 7.7: Itô’s formula8m

Week 7.8: Calculation of stochastic integrals using the Itô formula. Black-Scholes model6m

Week 7.9: Vasicek model. Application of the Itô formula to stochastic modelling5m

Week 7.10: Ornstein-Uhlenbeck process. Application of the Itô formula to stochastic modelling.4m

Stochastic integration12m

Week

8Upon 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....

10 videos (Total 94 min), 1 quiz

Week 8.2: Examples of Lévy processes. Calculation of the characteristic function in particular cases17m

Week 8.3: Relation to the infinitely divisible distributions7m

Week 8.4: Characteristic exponent8m

Week 8.5: Properties of a Lévy process, which directly follow from the existence of characteristic exponent7m

Week 8.6: Lévy-Khintchine representation and Lévy-Khintchine triplet-17m

Week 8.7: Lévy-Khintchine representation and Lévy-Khintchine triplet-27m

Week 8.8: Lévy-Khintchine representation and Lévy-Khintchine triplet-38m

Week 8.9: Modelling of jump-type dynamics. Lévy-based models7m

Week 8.10: Time-changed stochastic processes. Monroe theorem9m

Lévy processes12m

Week

9This module includes final exam covering all topics of this course...

1 quiz

Final Exam16m

4.4

32 Reviewsgot a tangible career benefit from this course

By SS•May 21st 2019

This course has less number of quiz questions but sufficient and well designed questions.

By ZM•Dec 1st 2018

Well presented course. I enjoyed it and was challenged a great deal. Thank you.

National Research University - Higher School of Economics (HSE) is one of the top research universities in Russia. Established in 1992 to promote new research and teaching in economics and related disciplines, it now offers programs at all levels of university education across an extraordinary range of fields of study including business, sociology, cultural studies, philosophy, political science, international relations, law, Asian studies, media and communicamathematics, engineering, and more.
Learn more on www.hse.ru...

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