Publication Date

5-2011

Advisor(s) - Committee Chair

Dr. Ferhan Atici (Director), Dr. Melanie Autin, Dr. Alex Lebedinsky

Degree Program

Department of Mathematics and Computer Science

Degree Type

Master of Science

Abstract

The concept of dynamic programming was originally used in late 1949, mostly during the 1950s, by Richard Bellman to describe decision making problems. By 1952, he refined this to the modern meaning, referring specifically to nesting smaller decision problems inside larger decisions. Also, the Bellman equation, one of the basic concepts in dynamic programming, is named after him. Dynamic programming has become an important argument which was used in various fields; such as, economics, finance, bioinformatics, aerospace, information theory, etc. Since Richard Bellman's invention of dynamic programming, economists and mathematicians have formulated and solved a huge variety of sequential decision making problems both in deterministic and stochastic cases; either finite or infinite time horizon. This thesis is comprised of five chapters where the major objective is to study both deterministic and stochastic dynamic programming models in finance.
In the first chapter, we give a brief history of dynamic programming and we introduce the essentials of theory. Unlike economists, who have analyzed the dynamic programming on discrete, that is, periodic and continuous time domains, we claim that trading is not a reasonably periodic or continuous act. Therefore, it is more accurate to demonstrate the dynamic programming on non-periodic time domains. In the second chapter we introduce time scales calculus. Moreover, since it is more realistic to analyze a decision maker’s behavior without risk aversion, we give basics of Stochastic Calculus in this chapter. After we introduce the necessary background, in the third chapter we construct the deterministic dynamic sequence problem on isolated time scales. Then we derive the corresponding Bellman equation for the sequence problem. We analyze the relation between solutions of the sequence problem and the Bellman equation through the principle of optimality. We give an example of the deterministic model in finance with all details of calculations by using guessing method, and we prove uniqueness and existence of the solution by using the Contraction Mapping Theorem. In the fourth chapter, we define the stochastic dynamic sequence problem on isolated time scales. Then we derive the corresponding stochastic Bellman equation. As in the deterministic case, we give an example in finance with the distributions of solutions.

Disciplines

Dynamical Systems | Finance and Financial Management