Course Summary
This course provides an introduction to stochastic processes, including Markov chains, Poisson processes, and Brownian motion, and their applications in various fields.Key Learning Points
- Learn the fundamentals of stochastic processes and their applications in various fields.
- Understand Markov chains, Poisson processes, and Brownian motion.
- Apply stochastic processes to real-world problems.
- Gain experience using software tools to simulate stochastic processes.
Related Topics for further study
Learning Outcomes
- Understand the basic concepts of stochastic processes, including Markov chains, Poisson processes, and Brownian motion.
- Be able to apply stochastic processes to real-world problems.
- Gain experience using software tools to simulate stochastic processes.
Prerequisites or good to have knowledge before taking this course
- Basic probability theory
- Calculus
Course Difficulty Level
IntermediateCourse Format
- Online self-paced
- Video lectures
- Assignments
- Quizzes
Similar Courses
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- Introduction to Probability
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Related Education Paths
Related Books
Description
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.
Outline
- Week 1: Introduction & Renewal processes
- About the University
- Welcome
- Week 1.1: Difference between deterministic and stochastic world
- Week 1.2: Difference between various fields of stochastics
- Week 1.3: Probability space
- Week 1.4: Definition of a stochastic function. Types of stochastic functions.
- Week 1.5: Trajectories and finite-dimensional distributions
- Week 1.6: Renewal process. Counting process
- Week 1.7: Convolution
- Week 1.8: Laplace transform. Calculation of an expectation of a counting process-1
- Week 1.9: Laplace transform. Calculation of an expectation of a counting process-2
- Week 1.10: Laplace transform. Calculation of an expectation of a counting process-3
- Week 1.11: Limit theorems for renewal processes
- About University
- Rules on the academic integrity in the course
- Applications of the Renewal Processes
- Quiz-1 answers and solutions
- Introduction & Renewal processes
- Week 2: Poisson Processes
- Week 2.1: 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-1
- 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-2
- 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-3
- 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-4
- Week 2.5: Memoryless property
- Week 2.6: Other definitions of Poisson processes-1
- Week 2.7: Other definitions of Poisson processes-2
- Week 2.8: Non-homogeneous Poisson processes-1
- Week 2.9: Non-homogeneous Poisson processes-2
- Week 2.10: Relation between renewal theory and non-homogeneous Poisson processes-1
- Week 2.11: Relation between renewal theory and non-homogeneous Poisson processes-2
- Week 2.12: Relation between renewal theory and non-homogeneous Poisson processes-3
- Week 2.13: Elements of the queueing theory. M/G/k systems-1
- Week 2.14: Elements of the queueing theory. M/G/k systems-2
- Week 2.15: Compound Poisson processes-1
- Week 2.16: Compound Poisson processes-2
- Week 2.17: Compound Poisson processes-3
- Applications of the Poisson Processes and Related Models
- Quiz-2 answers and solutions
- Poisson processes & Queueing theory
- Week 3: Markov Chains
- Week 3.1: Definition of a Markov chain. Some examples
- Week 3.2: Matrix representation of a Markov chain. Transition matrix. Chapman-Kolmogorov equation
- Week 3.3: Graphic representation. Classification of states-1
- Week 3.4: Graphic representation. Classification of states-2
- Week 3.5: Graphic representation. Classification of states-3
- Week 3.6: Ergodic chains. Ergodic theorem-1
- Week 3.7: Ergodic chains. Ergodic theorem-2
- Applications of the Markov chains
- Quiz-3 answers and solutions
- Markov Chains
- Week 4: Gaussian Processes
- Week 4.1: Random vector. Definition and main properties
- Week 4.2: Gaussian vector. Definition and main properties
- Week 4.3: Connection between independence of normal random variables and absence of correlation
- Week 4.4: Definition of a Gaussian process. Covariance function-1
- Week 4.5: Definition of a Gaussian process. Covariance function-2
- Week 4.6: Two definitions of a Brownian motion
- Week 4.7: Modification of a process. Kolmogorov continuity theorem
- Week 4.8: Main properties of Brownian motion
- Quiz-4 answers and solutions
- Gaussian processes
- Week 5: Stationarity and Linear filters
- Week 5.1: Two types of stationarity-1
- Week 5.2: Two types of stationarity-2
- Week 5.3: Spectral density of a wide-sense stationary process-1
- Week 5.4: Spectral density of a wide-sense stationary process-2
- Week 5.5: Stochastic integration of the simplest type
- Week 5.6: Moving-average filters-1
- Week 5.7: Moving-average filters-2
- Week 5.8: Moving-average filters-3
- Quiz-5 answers and solutions
- Stationarity and linear filters
- Week 6: Ergodicity, differentiability, continuity
- Week 6.1: Notion of ergodicity. Examples
- Week 6.2: Ergodicity of wide-sense stationary processes
- Week 6.3: Definition of a stochastic derivative
- Week 6.4: Continuity in the mean-squared sense
- Quiz-6 answers and solutions
- Ergodicity, differentiability, continuity
- Week 7: Stochastic integration & Itô formula
- Week 7.1: Different types of stochastic integrals. Integrals of the type ∫ X_t dt-1
- Week 7.2: Integrals of the type ∫ f(t) dW_t-1
- Week 7.3: Integrals of the type ∫ f(t) dW_t-2
- Week 7.4: Integrals of the type ∫ X_t dW_t-1
- Week 7.5: Integrals of the type ∫ X_t dW_t-2
- Week 7.6: Integrals of the type ∫ X_t dY_t, where Y_t is an Itô process
- Week 7.7: Itô’s formula
- Week 7.8: Calculation of stochastic integrals using the Itô formula. Black-Scholes model
- Week 7.9: Vasicek model. Application of the Itô formula to stochastic modelling
- Week 7.10: Ornstein-Uhlenbeck process. Application of the Itô formula to stochastic modelling.
- Quiz-7 answers and solutions
- Stochastic integration
- Week 8: Lévy processes
- Week 8.1: Definition of a Lévy process. Stochastic continuity and cà dlà g paths.
- Week 8.2: Examples of Lévy processes. Calculation of the characteristic function in particular cases
- Week 8.3: Relation to the infinitely divisible distributions
- Week 8.4: Characteristic exponent
- Week 8.5: Properties of a Lévy process, which directly follow from the existence of characteristic exponent
- Week 8.6: Lévy-Khintchine representation and Lévy-Khintchine triplet-1
- Week 8.7: Lévy-Khintchine representation and Lévy-Khintchine triplet-2
- Week 8.8: Lévy-Khintchine representation and Lévy-Khintchine triplet-3
- Week 8.9: Modelling of jump-type dynamics. Lévy-based models
- Week 8.10: Time-changed stochastic processes. Monroe theorem
- Truncation function in the Lévy-Khintchine representation
- Quiz-8 answers and solutions
- Lévy processes
- Final exam
- Final exam solution
- Final Exam
Summary of User Reviews
The Stochastic Processes course on Coursera has received positive reviews from many users. It is a comprehensive course that covers the basics of stochastic processes and their applications. One key aspect that many users thought was good is the clear and concise explanations provided by the instructor.Pros from User Reviews
- Clear and concise explanations
- Comprehensive coverage of basics
- Good for beginners
- Good examples and exercises
- Well-organized course structure
Cons from User Reviews
- Some users found the course too basic
- Some users found the course too theoretical
- Some users found the course difficult to follow
- No real-world applications covered
- No interaction with the instructor
Vladimir Panov
4.5 Raiting
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