## Elements of Probability Theory and Random Processes

- Details
- Parent Category: Курси
- Published: 27 May 2011

**Teaches: Shportyuk Volodymyr**

The course introduces the doctoral students to the instruments of probability theory and the theory of stochastic processes, which are necessary for understanding and developing mathematical models of derivatives and financial markets.

The main topics are: the basics of sets and mathematical logics; logics of predicates; expansion of sets of real numbers: complex numbers; expansion of the notion of a function. Maping. Metric and valuated spaces. Equivalent metrics and norms. Limits. Limit of a function. Main classes of sets. Rings, algebras, σ-rings, σ-algebras. Monotonic classes. Generated classes of sets. Set functions, measures, stochastic measures. Algebras generated by sets/events. Measurable sets. Indicators of sets. Algebras generated by random variables. Conditional expectations.

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**

- Angel de la Fuente
*Mathematical**Methods and Models for Economists*. Cambridge University Press, 2000. - Bobyk O.I. Beregova G.I., Kopytko B.I.
*Theory of Probabilities and Mathematical Statistics*. Kyiv, 2007. - Dorogovtsev A.Y.
*Elements of a General Theory of Limits and Integral*. Kyiv: Vushcha Shkola, 1989. - Sheftel Z.G.
*Theory of Probabilities*. Kyiv: Vyshcha Shkola, 1994 - 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 - 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 - Karlin, H. M. Taylor.
*A First Course in Stochastic Processes*. 2^{nd}ed. Academic Press, 1975. - Karlin, H. M. Taylor.
*A Second Course in Stochastic Processes*. Academic Press, 1981. - Karlin, H. M. Taylor.
*An Introduction to Stochasic Modeling*, Third Edition. Academic Press, 1998. - Karlin.
*Mathematical Methods and Theory in Games, Programming, and Economics*. Dover Publications, 1992. ISBN 978-0486670201