Mathematics for economics

Lecturer:

Tutor:

Supervisor:

Department of Mathematics

Synopsis:

Aim of this subject is to give the basic informations about probability, mathematical statistics, random processes, especially Markov chains, and to show their applications, mainly in financial and insurance mathematics. At the end of the course, bases of cluster analysis will be shown.

Requirements:

The details are at http://math.feld.cvut.cz/helisova/mekX01MEK.html

Syllabus of lectures:

1. Random sample, statistics and their distributions.

2. Point estimators of distribution parameters and their properties, method of moments and the method of maximum likehood.

3. Confidence intervals for parameters a construction of tests of hypotheses on parameters.

4. Tests of hypotheses on mean value and variance of the normal distribution.

5. ChiSquare-tests for goodness of fit.

6. Random processes - fundamental definitions.

7. Markov chains with discrete time - basic properties, random walk, transition matrix,

Chapman-Kolmogorov equation, states classification.

8. Markov chains with continuous time - Wiener process, Poisson process.

9. Stochastic integral, stochastic differential equations, Black-Scholes formula.

10. General insurance - basic probability distributions of the number of events and claim amounts.

11. Technical reserves - indemnity reserve, triangular schemes. Markov chains in bonus systems.

12. Life insurance - premium in capital and annuity insurance.

13. Cluster analysis - basic definitions.

14. Basic methods of clustering.

Syllabus of tutorials:

1. Probability of random event, conditional probability, Bayes theorem.

2. Discrete random variable - distribution function, expected value, variance.

3. Continuous random variable - density, distribution function, expected value, variance.

4. Central limit theorem.

5. Point and interval parameters estimates.

6. Hypotheses testing.

7. Random processes - states classification.

8. Markov chains with discrete time - transition matrix.

9. Markov chains with continuous time - Wiener process, Poisson process.

10. Stochastic differential equations.

11. Calculation of premium and reserves in general insurance.

12. Calculation of premium in capital and annuity insurance.

13. Basic methods of clustering.

14. Backup

Study Objective:

Study materials:

1. Grinstead, Ch.M., Snell, J. L.: Introduction to Probability. American Math. Society, 1997.

2. Ross, S.M.: Stochastic Processes. John Wiley & Sons, 1982.

3. Kaas, R., Goovaerts, M., Dhaene, J., Denuit, M.: Modern actuarial risk theory. Kluwer Academic Publishers, 2004.

4. Gerber, H.U.: Life Insurance Mathematics. Springer-Verlag, New York-Berlin-Heidelberg, 1990.

5. Duda, R.O., Hart, P.E., Stork, D.G.: Pattern Classification. John Wiley & Sons, 2001.

Note:

Further information:

No time-table has been prepared for this course

The course is a part of the following study plans:

• Economics and Management of Power Engineering- structured studies (compulsory course)
• Electronics - Electronic Applications- structured studies (compulsory course)