Financial Mathematics

Teaches: Shportyuk Volodymyr

Course Schedule

Theme 1. Fundamentals of the Theory of Sets and Mathematical Logics. Logics of Predicates. Complex Numerals as The Broadening of Real Numbers.

Theme 2. Broadening of the Function Concept. Mapping. Metric and normalized Space. Equivalent Metrics and Norms. Convergence. Limit of a Function.

Theme 3. Main Classes of Sets. Rings, Algebras. Σ-rings, Σ-algebras. Monotonous Classes. Resulted Classes of Sets.

Theme 4. Functions of Sets and Limits. Probability Limits.

Theme 5. Functions of Random Argument.

Theme 6. Algebras Resulted From Sets or Events. Measurable Sets. Indicators of Sets. Algebras Resulted from Random Events. Conditional Mathematical Expectations.

Theme 7. Discrete Multivariable Random Events.

Theme 8. Continuous Multivariable Random Events.

Theme 9. Random Processes. Markov Processes with Discrete Time and Discrete Status.

Theme 10.Ergodic Markov’s Chains.

Theme 11. Markov Processes with Continuous Status. Kolmogorov-Chapmen System of Equations.

Literature

1. Angel de la Fuente Mathematical Methods and Models for Economists Cambridge University Press, 2000.
2. Bobyk O.I. Beregova G.I., Kopytko B.I. Theory of Probabilities and Mathematical Statistics. Kyiv, 2007.
3. Dorogovtsev A.Y. Elements of a General Theory of Limits and Integral. Kyiv: Vushcha Shkola, 1989.
4. Efe A. Ok Real Analysis with Economic Applications., Princeton University Press, 2007, 664 pp.
5. Efe A. Ok Probability Theory with Economic Applications (http://homepages.nyu.edu/~eo1/books.html).
6. Karlin and H. M. Taylor. A First Course in Stochastic Processes. 2^nd ed. Academic Press, 1975.
7. Karlin and H. M. Taylor. A Second Course in Stochastic Processes. Academic Press, 1981.
8. Karlin and H. M. Taylor. An Introduction to Stochastic Modeling, Third Edition. Academic Press, 1998.
9. Karlin. Mathematical Methods and Theory in Games, Programming, and Economics. Dover Publications, 1992. ISBN 978-0486670201.
10. Sheftel Z.G. Theory of Probabilities. Kyiv: Vyshcha Shkola, 1994
11. Shreve, Steven E. Stochastic Calculus for Finance I. The Binomial Asset Pricing Model Series: Springer Finance 2005, XVI, 187 p. ISBN: 978-0-387-24968-1
12. Shreve, Steven E. Stochastic Calculus for Finance II. Continuous-Time Models Series: Springer Finance. 2nd printing, 2004, XIX, 550 p., ISBN: 978-0-387-40101-0
13. Sokolov G.A., Chistiakova N.A. Controlled Markov’s Chains in Economics. Moscow: PH Fizmathlit, 2005.

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