National Research University - Higher School of Economics
Course Title “Economic and Mathematic Modeling”
Master’s Program 38.04.05 “Big Data Systems”
The Government of the Russian Federation
The Federal State Autonomous Institution of Higher Education
"National Research University - Higher School of Economics"
Faculty of Business Informatics
Department of Innovation and Business in Information Technology
Course Title “Economic an Mathematic Modeling”
Master’s Program 38.04.05 “Big Data Systems”
Author:
Dr. Sci., Prof. Andrey Dmitriev, a.dmitriev@hse.ru
Approved by the meeting of the Department
Head of Department
Prof. Svetlana Maltseva
Moscow, 2014
This document may not be reproduced or redistributed by other Departments of the University without
permission of the Authors.
National Research University - Higher School of Economics
Course Title “Economic and Mathematic Modeling”
Master’s Program 38.04.05 “Big Data Systems”
Field of Application and Regulations
The course "Economic and Mathematic Modeling" syllabus lays down minimum requirements for student’s knowledge and skills; it also provides description of both contents and forms of training and assessment in use. The course is offered to students of the Master’s Program "Big Data Systems" (area code
080500.68) in the Faculty of Business Informatics of the National Research University "Higher School of
Economics". The course is a part of the curriculum pool of required courses (1st year, M.1.Б Core courses,
M.1 Courses required by the standard 38.04.05 of the 2014-2015 academic year’s curriculum), and it is a
one-module course (1st module). The duration of the course amounts to 32 class periods (both lecture and
seminars) divided into 16 lecture hours and 16 practice hours. Besides, 76 academic hours are set aside to
students for self-studying activity.
The syllabus is prepared for teachers responsible for the course (or closely related disciplines),
teaching assistants, students enrolled on the course "Economic and Mathematic Modeling" as well as experts and statutory bodies carrying out assigned or regular accreditations in accordance with
 educational standards of the National Research University – Higher School of Economics,
 curriculum ("Business Informatics", area code 38.04.05), Big Data Systems specialization, 1st year,
2014-2015 academic year.
1
Course Objectives
The main objective of the Course is to present, examine and discuss with students fundamentals and principles of economic and mathematic modeling. This course is focused on understanding the role of mathematic modeling for quantitative analysis of stochastic and dynamic economic systems.
Generally, the objective of the course can be thought as a combination of the following constituents:
 familiarity with peculiarities of bifurcation theory, catastrophe theory, chaos theory, Levy random
walk and minority games theory as applied areas related to economic and mathematic modeling,
 understanding of the main notions of dynamic and stochastic systems theory; the framework of dynamic and stochastic modeling as the most significant areas of economic systems studies, understanding of the main notions of dynamic and stochastic systems theory; the framework of dynamic
and stochastic modeling as the most significant areas of economic systems studies,
 understanding of the role of mathematic modeling in financial and economic modeling,
 obtaining skills in utilizing nonlinear dynamic modeling in economic problem solving,
 obtaining skills in utilizing stochastic modeling in financial problem solving,
 understanding of the role of equilibrium theory and instabilities in economic modeling,
 understanding of the role of stable distributions in financial process modeling.
2
Students' Competencies to be Developed by the Course
While mastering the course material, the student will
 know main notions of the bifurcation theory, catastrophe theory, chaos theory, random walk theory
and stochastic process theory,
 acquire skills of analyzing and solving economic and mathematic problems,
 gain experience in economic and mathematic modeling with use main notions of the bifurcation
theory, catastrophe theory, chaos theory, random walk theory and stochastic process theory.
In short, the course contributes to the development of the following professional competencies:
Ccompetencies
Ability to offer concepts,
models, invent and test
FSES/
HSE
code
Descriptors – main mastering
features (indicators of result
achievement)
SC-2
Demonstrates
Training forms and
methods contributing
to the formation and
development of
competence
Lecture, practice, homeworks
National Research University - Higher School of Economics
Course Title “Economic and Mathematic Modeling”
Master’s Program 38.04.05 “Big Data Systems”
Ccompetencies
methods and tools for professional work
Ability to apply the methods
of system analysis and modeling to assess, design and
strategy development of enterprise architecture
Ability to develop and implement economic and mathematical models to justify
the project solu-tions in the
field of information and
computer technology
Ability to organize self and
collective research work in
the enterprise and manage it
3
Training forms and
methods contributing
to the formation and
development of
competence
FSES/
HSE
code
Descriptors – main mastering
features (indicators of result
achievement)
PC-13
Owns and uses
Lecture, practice, homeworks
PC-14
Owns and uses
Lecture, practice, homeworks
PC-16
Demonstrates
Lecture, practice, homeworks
The Course within the Program’s Framework
The course "Economic and Mathematic Modeling" syllabus lays down minimum requirements for student’s knowledge and skills; it also provides description of both contents and forms of training and assessment in use. The course is offered to students of the Master’s Program "Big Data Systems" (area code
080500.68) in the Faculty of Business Informatics of the National Research University "Higher School of
Economics". The course is a part of the curriculum pool of required courses (1st year, M.1.Б Core courses,
M.1 Courses required by the standard 080500.68 of the 2014-2015 academic year’s curriculum), and it is a
one-module course (1st module). The duration of the course amounts to 32 class periods (both lecture and
seminars) divided into 16 lecture hours and 16 practice hours. Besides, 76 academic hours are set aside to
students for self-studying activity.
Academic control forms include
 1 class assignment, which implies problems solving in the end of 1st module; material to be covered
by class assignment is fully determined by both course schedule and topics discussed by the corresponding date,
 8 homeworks are done by students individually, herewith each student has to prepare electronic
(PDF format solely) report; all reports have to be submitted in LMS; all reports are checked and
graded by the instructor on ten-point scale by the end of the 1st module,
 pass-final examination, which implies written test and computer-based problem solving.
The Course is to be based on the acquisition of the following courses:
 Calculus
 Linear Algebra
 Probability Theory and Mathematical Statistics
 Macroeconomics
 Microeconomics
The Course requires the following students' competencies and knowledge:
 main definitions, theorems and properties from Calculus, Linear Algebra, Probability Theory and
Mathematical Statistics, Macroeconomics and Microeconomics courses,
 ability to communicate both orally and in written form in English language,
National Research University - Higher School of Economics
Course Title “Economic and Mathematic Modeling”
Master’s Program 38.04.05 “Big Data Systems”
 ability to search for, process and analyze information from a variety of sources.
Main provisions of the course should be used to further the study of the following courses:
 Advanced Data Analysis and Big Data for Business
 Predictive Modeling
 Applied Machine Learning
4
Thematic Course Contents
№
1
2
3
4
5
6
7
8
Class hours
Hours
(total
number)
Title of the topic / lecture
Economical dynamics, growth and equilibrium
Systems of first-order differential equations
and economic dynamics
Bifurcation theory and economic dynamics
Catastrophe theory and economic dynamics
Chaos theory and economic dynamics
Probability distribution function of stock
market instruments. Correlation in the stock
market
Random walk in financial market models
Minority games
TOTAL
5
Practice
Independent work
2
2
8
2
2
8
2
2
2
2
2
2
2
2
12
12
12
8
2
2
16
2
2
16
8
8
76
Lectures
Seminars
Forms and Types of Testing
Type of
control
Current
(week)
Form of control
Class assignment
Homework
Resultant Pass-fail exam
1
week 8
week 1 to
week 8
week 9
1 year
2
3
Department
Parameters
Innovation
and Business in Information
Technology
problems solving, written
report (paper) – 120 minute
LMS electronic report
4
written test (paper) and
computer-based problem
solving
Evaluation Criteria
Current and resultant grades are made up of the following components:
 1 class assignment
implies problems solving in the end of 1st module; material to be covered by class assignment is fully determined by both course schedule and topics discussed by the corresponding date. The class assignment
(CA) is assessed on the ten-point scale.
 8 homeworks
are done by students individually, herewith each student has to prepare electronic (PDF format solely) report. All reports have to be submitted in LMS. All reports are checked and graded by the instructor on tenpoint scale by the end of the 1st module. All homeworks (HW) is assessed on the ten-point scale summary.
 pass-final examination
implies written test (WT) and computer-based problem solving (CS).
Finally, the total course grade on ten-point scale is obtained as
National Research University - Higher School of Economics
Course Title “Economic and Mathematic Modeling”
Master’s Program 38.04.05 “Big Data Systems”
O(Total) = 0,2 * O(HW) + 0,4 * O(CA) + 0,1 * O(WT) + 0,3 * O(CS).
A grade of 4 or higher means successful completion of the course ("pass"), while grade of 3 or lower means unsuccessful result ("fail"). Conversion of the concluding rounded grade O(Total) to five-point
scale grade.
6
Detailed Course Contents
Lecture 1. Economical dynamics, growth and equilibrium
Lecture’s content:
 Solow-Swan model. Assumptions of the model. Mathematics of the model. Balanced-growth equilibrium. Golden rule. Production function. Dynamical system. Stability of the dynamical system:
Lyapunov stability, asymptotic stability, orbital stability. Dynamic equilibrium. Koopmans theorem
(existence of an equilibrium in the Solow model). Arrow-Gurwicz theorem (existence of stable
equilibrium). Asymptotic stability of the model.
 Attractors and repellers.
 Walrasian and Marshallian equilibriums. Open and closed systems. Dynamic and static equilibrium.
Practice 1. Ramsey-Cass-Koopmans model. Problem of consumer choice.
Pontryagin’s maximum principle. General economic equilibrium. Modified golden rule.
Attractor of Ramsey-Cass-Koopmans model.
Materials required
1. Zang W.B. synergetic Economics. Time and Change in Nonlinear Economics. Springer, 1991.
Recommended readings
1. Shone R. Economic Dynamic. Phase Diagrams and Their Economic Applications. 2nd Edition.
Cambridge Press, 2002
2. Zang W.B. Differential Equations, Bifurcations, and Chaos in Economics. World Scientific Publishing Company, 2005.
3. Puu T. Attractors, Bifurcations, and Chaos: Nonlinear Phenomena in Economics, 2nd Edition,
Springer, 2003.
4. Rosser J.B. From Catastrophe to Chaos: A General Theory of Economic Discontinuities. 2nd Edition, Springer, 2000.
5. Stachurski D. Economic Dynamics: Theory and Computation. MIT Press, 2009.
Lecture 2. Systems of first-order differential equations and economic dynamics
Lecture’s content:
 Goodwin model. Lotka-Volterra equation. Equilibrium points. Linearization. Jacobian matrix. Eigenvalues of the matrix. Equilibrium points of linear autonomous system: saddle point, stable node,
unstable node, stable focus, unstable focus, centre. Phase diagram. The principle of linearized stability. Equilibrium points of Lotka-Volterra equation. Structural stability. Structural instability of
Lotka-Volterra model. Conservative system. Conservative Lotka-Volterra system.
 Slow and fast variables. Tikhonov theorem on dynamical systems.
Practice 2. The simplest model of competition between two firms. Competition
model with a limited production growth. Bazykin model. Mankiw-Romer-Weil model.
Accounting for external.
National Research University - Higher School of Economics
Course Title “Economic and Mathematic Modeling”
Master’s Program 38.04.05 “Big Data Systems”
Materials required
1. Zang W.B. synergetic Economics. Time and Change in Nonlinear Economics. Springer, 1991.
Recommended readings
1. Shone R. Economic Dynamic. Phase Diagrams and Their Economic Applications. 2nd Edition.
Cambridge Press, 2002
2. Zang W.B. Differential Equations, Bifurcations, and Chaos in Economics. World Scientific Publishing Company, 2005.
3. Puu T. Attractors, Bifurcations, and Chaos: Nonlinear Phenomena in Economics, 2nd Edition,
Springer, 2003.
4. Rosser J.B. From Catastrophe to Chaos: A General Theory of Economic Discontinuities. 2nd Edition, Springer, 2000.
5. Stachurski D. Economic Dynamics: Theory and Computation. MIT Press, 2009.
Lecture 3. Bifurcation theory and economic dynamics
Lecture’s content:
Modeling regional dynamics. Local bifurcations: saddle-node (fold) bifurcation, transcritical bifurcation, Pitchfork bifurcation, period-doubling (flip) bifurcation, Hopf bifurcation, Neimark bifurcation.
 Global bifurcations: homoclinic bifurcation, heteroclinic bifurcation, infinite-period bifurcation,
blue sky catastrophe. Codimension of a bifurcation. Bifurcation diagram. Flows. Hopf theorem. The
Poincare-Bendixson theorem.
 Limit cycle (attractor). Stable, unstable and semi-stable limit cycle.

Practice 3. Dynamic transportation modal choice. Oscillations in van der Ploeg’s
hybrid growth model. Periodic optimal employment policy. Optimal economic growth
associated with endogenous fluctuations.
Materials required
1. Zang W.B. synergetic Economics. Time and Change in Nonlinear Economics. Springer, 1991.
Recommended readings
1. Shone R. Economic Dynamic. Phase Diagrams and Their Economic Applications. 2nd Edition.
Cambridge Press, 2002
2. Zang W.B. Differential Equations, Bifurcations, and Chaos in Economics. World Scientific Publishing Company, 2005.
3. Puu T. Attractors, Bifurcations, and Chaos: Nonlinear Phenomena in Economics, 2nd Edition,
Springer, 2003.
4. Rosser J.B. From Catastrophe to Chaos: A General Theory of Economic Discontinuities. 2nd Edition, Springer, 2000.
5. Stachurski D. Economic Dynamics: Theory and Computation. MIT Press, 2009.
Lecture 4. Catastrophe theory and economic dynamics
Lecture’s content:
 Business cycles in the Kaldor model. Structural stability. Morse lemma. Thom theorem. Morse critical points. Degenerate critical points. Thom elementary catastrophes.
 Potential functions of one active variable: fold catastrophe, cusp catastrophe, Swallowtail catastrophe, butterfly catastrophe. Potential functions of two active variables: hyperbolic umbilic catastrophe, elliptice umbilic catastrophe, parabolic umbilic catastrophe.
National Research University - Higher School of Economics
Course Title “Economic and Mathematic Modeling”
Master’s Program 38.04.05 “Big Data Systems”
Practice 4. Resource management. Multiple equilibria in Wilson’s retail model.
Stock market forecasting.
Materials required
1. Zang W.B. synergetic Economics. Time and Change in Nonlinear Economics. Springer, 1991.
Recommended readings
1. Shone R. Economic Dynamic. Phase Diagrams and Their Economic Applications. 2nd Edition.
Cambridge Press, 2002
2. Zang W.B. Differential Equations, Bifurcations, and Chaos in Economics. World Scientific Publishing Company, 2005.
3. Puu T. Attractors, Bifurcations, and Chaos: Nonlinear Phenomena in Economics, 2nd Edition,
Springer, 2003.
4. Rosser J.B. From Catastrophe to Chaos: A General Theory of Economic Discontinuities. 2nd Edition, Springer, 2000.
5. Stachurski D. Economic Dynamics: Theory and Computation. MIT Press, 2009.
Lecture 5. Chaos theory and economic dynamics
Lecture’s content:
 Chaotic dynamic price formation. Lorenz system. Properties of the Lorenz system: homogeneity,
symmetry, dissipativity, bounded trajectories. Equilibrium points of the Lorenz system. Lyapunov
stability of equilibrium points. Lorenz attractor. Lorenz map.
 Measures of chaos: Lyapunov exponent, correlation function, Hausdorff-Besicovitch fractal dimension, Renyi fractal dimension. Fractal. Sensitivity to initial conditions. Strange attractors.
 Chaos and economic forecasting. Deterministic systems and time series.
Practice 5. Chaotic dynamic of cities. Chaos in an international economic model.
Materials required
1. Zang W.B. synergetic Economics. Time and Change in Nonlinear Economics. Springer, 1991.
Recommended readings
1. Shone R. Economic Dynamic. Phase Diagrams and Their Economic Applications. 2nd Edition.
Cambridge Press, 2002
2. Zang W.B. Differential Equations, Bifurcations, and Chaos in Economics. World Scientific Publishing Company, 2005.
3. Puu T. Attractors, Bifurcations, and Chaos: Nonlinear Phenomena in Economics, 2nd Edition,
Springer, 2003.
4. Rosser J.B. From Catastrophe to Chaos: A General Theory of Economic Discontinuities. 2nd Edition, Springer, 2000.
5. Stachurski D. Economic Dynamics: Theory and Computation. MIT Press, 2009.
Lecture 6. Probability distribution function of stock market instruments.
Correlation in the stock market

Lecture’s content:
Empirical distributions of stock returns and number of shares. Stable distribution. Characteristic
function. Properties of stable distribution: infinitely divisible, leptokurtotic, closure under convolution. A generalized central limit theorem.
National Research University - Higher School of Economics
Course Title “Economic and Mathematic Modeling”
Master’s Program 38.04.05 “Big Data Systems”

Levy distribution. Probability density function and characteristic function of the Levy distribution.
Autocorrelation function and spectral density. Higher-order correlation: the volatility. Stationarity
of price changes.
Practice 6. Empirical distributions of share volumes. Empirical distributions of
the number of transactions. Empirical distributions of time interval between transactions.
Materials required
1. Mantegna R.N., Stanley H.E. An Introduction to Econophysics, Correlation and Complexity in
Finance. Cambridge University Press, 2000.
Recommended readings
1. Malliaris A.G., Brock W.A. Stochastic Methods in Economics and Finance. North Holland,
1988.
2. Rachev S.T., Kim Y.S. Financial Models with Levy processes and Volatility Clustering. Wiley,
2011.
3. Schoutens W. Levy Processes in Finance: Pricing Financial Derivatives. Wiley, 2003.
4. Bottasso A. Stable Levy Processes in Finance: Economics. Lambert, 2011.
5. Challet D., Marsili M. Minority Games: Interacting Agents in Financial Markets. Oxford Press,
2013
Lecture 7. Random walk in financial market models
Lecture’s content:
Options pricing. Lattice random walk. One-dimensional random walk. Gaussian random walk. The
speed of convergence. Berry-Essen theorem 1. Berry-Essen theorem 2. Basin of attraction.
 Levy flight. Mandelbrot survival function. Scale invariant. Truncated Levy flights and fluctuations
of stock market instruments. Functional Levy flight.
 Holtsmark distribution.

Practice 7. Black-Scholes equation.
Materials required
1. Mantegna R.N., Stanley H.E. An Introduction to Econophysics, Correlation and Complexity in
Finance. Cambridge University Press, 2000.
Recommended readings
1. Malliaris A.G., Brock W.A. Stochastic Methods in Economics and Finance. North Holland,
1988.
2. Rachev S.T., Kim Y.S. Financial Models with Levy processes and Volatility Clustering. Wiley,
2011.
3. Schoutens W. Levy Processes in Finance: Pricing Financial Derivatives. Wiley, 2003.
4. Bottasso A. Stable Levy Processes in Finance: Economics. Lambert, 2011.
5. Challet D., Marsili M. Minority Games: Interacting Agents in Financial Markets. Oxford Press,
2013
Lecture 8. Minority games
Lecture’s content:
Price dynamics. Formulation of the minority game. Thermal minority game. Minority game without information. Grand-canonical minority game. Analytic approach.
National Research University - Higher School of Economics
Course Title “Economic and Mathematic Modeling”
Master’s Program 38.04.05 “Big Data Systems”
Practice 8. Speculative trading.
Materials required
1. Challet D., Marsili M. Minority Games: Interacting Agents in Financial Markets. Oxford Press,
2013
Recommended readings
1. Malliaris A.G., Brock W.A. Stochastic Methods in Economics and Finance. North Holland,
1988.
2. Rachev S.T., Kim Y.S. Financial Models with Levy processes and Volatility Clustering. Wiley,
2011.
3. Schoutens W. Levy Processes in Finance: Pricing Financial Derivatives. Wiley, 2003.
4. Bottasso A. Stable Levy Processes in Finance: Economics. Lambert, 2011.
7
Educational Technology
During classes various types of active methods are used: analysis of practical problems, group
work, computer simulations in computational software program Mathematica, distance learning with use
LMS.
8
8.1
Methods and Materials for Current Control and Attestation
Example of Problems for Class Assignment
Problem 1. Solve the nonhomogeneous differential equation
𝑑𝜋
= 𝛽𝑓(𝑢) − 𝛽(1 − 𝜉)𝜋
𝑑𝑡
for 𝜋(0) = 𝜋0 .
Show that for 𝛽 > 0 and 0 < 𝜉 < 1 the equilibrium 𝜋 ∗ is asymptotically stable.
Problem 2. Given the following parameters for the Solow growth model
𝑎 = 4, 𝛼 = 0.25, 𝑠 = 0.1, 𝛿 = 0.4, 𝑛 = 0.03
(i) to plot the graph of 𝑘(𝑡)
(ii) plot the function 𝑘̇ = 𝑠𝑎𝑘 𝛼 − (𝑛 + 𝛿)𝑘
(iii) Linearise 𝑘̇ about the equilibrium in (ii) and establish whether it is stable or unstable.
Problem 3. For the following Holling–Tanner predatory–prey model
𝑥
6𝑥𝑦
𝑥̇ = 𝑥 (1 − ) −
6
8 + 8𝑥
0.4𝑦
𝑦̇ = 0.2𝑦 (1 −
)
𝑥
(i) Find the fixed points.
(ii) Do any of the fixed points exhibit a stable limit cycle?
Problem 4. The value of a share at time is t
𝑋(𝑡) = 𝑥0 + 𝐷(𝑡),
where 𝑥0 > 0 and [𝐷(𝑡), 𝑡 ≥ 0] is a Brownian motion with positive drift parameter μ and variance parameter σ2. At time point t=0 a speculator acquires an American call option on this share with finite expiry date
τ. Assume that
𝑥0 + 𝜇𝑡 > 3𝜎√𝑡, 0 ≤ 𝑡 ≤ 𝜏.
(i) Why does the assumption make sense?
National Research University - Higher School of Economics
Course Title “Economic and Mathematic Modeling”
Master’s Program 38.04.05 “Big Data Systems”
(ii) When should the speculator exercise to make maximal mean undiscounted profit?
8.2
Questions for Pass-Final Examination
Economic Models
1. Solow-Swan model. Assumptions of the model. Mathematics of the model. Balanced-growth equilibrium. Golden rule. Production function.
2. Ramsey-Cass-Koopmans model. Problem of consumer choice. Pontryagin’s maximum principle.
General economic equilibrium. Modified golden rule.
3. Goodwin model.
4. The simplest model of competition between two firms.
5. Competition model with a limited production growth.
6. Bazykin model.
7. Mankiw-Romer-Weil model. Accounting for external.
8. Modeling regional dynamics.
9. Dynamic transportation modal choice.
10. Oscillations in van der Ploeg’s hybrid growth model.
11. Periodic optimal employment policy.
12. Optimal economic growth associated with endogenous fluctuations.
13. Business cycles in the Kaldor model.
14. Resource management.
15. Multiple equilibria in Wilson’s retail model.
16. Stock market forecasting.
17. Chaotic dynamic price formation.
18. Chaotic dynamic of cities.
19. Chaos in an international economic model.
20. Empirical distributions of stock returns and number of shares.
21. Empirical distributions of share volumes.
22. Empirical distributions of the number of transactions.
23. Empirical distributions of time interval between transactions.
24. Options pricing.
25. Black-Scholes equation.
26. Price dynamics.
27. Speculative trading.
Mathematical Foundations
1. Dynamical system. Stability of the dynamical system: Lyapunov stability, asymptotic stability, orbital stability.
2. Dynamic equilibrium. Koopmans theorem (existence of an equilibrium in the Solow model).
3. Arrow-Gurwicz theorem (existence of stable equilibrium). Asymptotic stability of the model.
4. Attractor of Ramsey-Cass-Koopmans model.
5. Attractors and repellers.
6. Open and closed systems. Dynamic and static equilibrium.
National Research University - Higher School of Economics
Course Title “Economic and Mathematic Modeling”
Master’s Program 38.04.05 “Big Data Systems”
7. Lotka-Volterra equation. Equilibrium points. Linearization. Jacobian matrix. Eigenvalues of the
matrix. Equilibrium points of linear autonomous system: saddle point, stable node, unstable node,
stable focus, unstable focus, centre. Phase diagram.
8. The principle of linearized stability. Equilibrium points of Lotka-Volterra equation. Structural stability. Structural instability of Lotka-Volterra model. Conservative system. Conservative LotkaVolterra system. Attrractors. Slow and fast variables.
9. Tikhonov theorem on dynamical systems.
10. Local bifurcations: saddle-node (fold) bifurcation, transcritical bifurcation, Pitchfork bifurcation,
period-doubling (flip) bifurcation, Hopf bifurcation, Neimark bifurcation.
11. Global bifurcations: homoclinic bifurcation, heteroclinic bifurcation, infinite-period bifurcation,
blue sky catastrophe. Codimension of a bifurcation. Bifurcation diagram. Flows. Hopf theorem. The
Poincare-Bendixson theorem.
12. Limit cycle (attractor). Stable, unstable and semi-stable limit cycle.
13. Structural stability. Morse lemma. Thom theorem. Morse critical points. Degenerate critical points.
Thom elementary catastrophes.
14. Potential functions of one active variable: fold catastrophe, cusp catastrophe, Swallowtail catastrophe, butterfly catastrophe.
15. Potential functions of two active variables: hyperbolic umbilic catastrophe, elliptice umbilic catastrophe, parabolic umbilic catastrophe.
16. Lorenz system. Properties of the Lorenz system: homogeneity, symmetry, dissipativity, bounded
trajectories.
17. Equilibrium points of the Lorenz system. Lyapunov stability of equilibrium points. Lorenz attractor.
Lorenz map.
18. Measures of chaos: Lyapunov exponent, correlation function, Hausdorff-Besicovitch fractal dimension, Renyi fractal dimension.
19. Fractal. Sensitivity to initial conditions. Strange attractors.
20. Chaos and economic forecasting.
21. Deterministic systems and time series.
22. Stable distribution. Characteristic function. Properties of stable distribution: infinitely divisible, leptokurtotic, closure under convolution.
23. A generalized central limit theorem.
24. Levy distribution. Probability density function and characteristic function of the Levy distribution.
25. Autocorrelation function and spectral density. Higher-order correlation: the volatility.
26. Lattice random walk. One-dimensional random walk. Gaussian random walk. The speed of convergence.
27. Berry-Essen theorem 1. Berry-Essen theorem 2.
28. Basin of attraction.
29. Levy flight.
30. Mandelbrot survival function. Scale invariant.
31. Truncated Levy flights and fluctuations of stock market instruments.
32. Functional Levy flight.
33. Holtsmark distribution.
34. Formulation of the minority game.
35. Thermal minority game.
National Research University - Higher School of Economics
Course Title “Economic and Mathematic Modeling”
Master’s Program 38.04.05 “Big Data Systems”
36. Minority game without information.
37. Grand-canonical minority game.
Example of Problem
Consider the following Walrasian price and quantity adjustment model
𝜙 ′ (𝑌) = 0.5 + 0.25𝑌
𝐷(𝑝) = −0.025𝑝3 + 0.75𝑝2 − 6𝑝 + 40
𝑝̇ = 0.75[𝐷(𝑝, 𝜙(𝑌)) − 𝑌]
𝑌̇ = 2[𝑝 − 𝜙′(𝑌)]
(i) What is the economically meaningful fixed point of this system?
(ii) Does this system have a stable limit cycle?
9
Teaching Methods and Information Provision
9.1 Core Textbook
Mantegna R.N., Stanley H.E. An Introduction to Econophysics, Correlation and Complexity in Finance.
Cambridge University Press, 2000.
Zang W.B. Synergetic Economics. Time and Change in Nonlinear Economics. Springer, 1991
Challet D., Marsili M. Minority Games: Interacting Agents in Financial Markets. Oxford Press, 2013
9.2 Required Reading
Shone R. Economic Dynamic. Phase Diagrams and Their Economic Applications. 2nd Edition. Cambridge
Press, 2002
Zhang W.B. Differential Equations, Bifurcations, and Chaos in Economics. World Scientific Publishing
Company, 2005.
Malliaris A.G., Brock W.A. Stochastic Methods in Economics and Finance. North Holland, 1988.
Rachev S.T., Kim Y.S. Financial Models with Levy processes and Volatility Clustering. Wiley, 2011.
9.3 Supplementary Reading
Puu T. Attractors, Bifurcations, and Chaos: Nonlinear Phenomena in Economics, 2nd Edition, Springer,
2003.
Rosser J.B. From Catastrophe to Chaos: A General Theory of Economic Discontinuities. 2nd Edition,
Springer, 2000.
Stachurski D. Economic Dynamics: Theory and Computation. MIT Press, 2009.
Schoutens W. Levy Processes in Finance: Pricing Financial Derivatives. Wiley, 2003.
Bottasso A. Stable Levy Processes in Finance: Economics. Lambert, 2011.
Challet D., Marsili M. Minority Games: Interacting Agents in Financial Markets. Oxford Press, 2013
Ian Jacques. Mathematics for Economics and Business. Pearson, 7 edition, 2012.
Kenneth Shaw. Mathematical Modeling in Business and Economics: A Data-Driven Approach. Business
Expert Press, September, 2014
9.4 Handbooks
Gardiner C. Stochastic Methods: A Handbook for the Natural and Social Sciences. Springer, 2009
Schemedders K., Judd K. Handbook of Computational Economics. V.3. Elsevier, 2014
9.5 Software
Mathematica v. 9.0
National Research University - Higher School of Economics
Course Title “Economic and Mathematic Modeling”
Master’s Program 38.04.05 “Big Data Systems”
9.6 Distance Learning
HSE Electronic Library access: Books24*7, Scopus, EBSCOHost, Science Direct, Web of Knowledge
HSE Learning Management System
10 Technical Provision
Computer, projector (for lectures or practice), computer class