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Über dieses Buch

This open access textbook is the first to provide Business and Economics Ph.D. students with a precise and intuitive introduction to the formal backgrounds of modern financial theory. It explains Brownian motion, random processes, measures, and Lebesgue integrals intuitively, but without sacrificing the necessary mathematical formalism, making them accessible for readers with little or no previous knowledge of the field. It also includes mathematical definitions and the hidden stories behind the terms discussing why the theories are presented in specific ways.

Inhaltsverzeichnis

Frontmatter

Open Access

1. Introduction

Abstract
Anyone who is occupied with modern financing theory will soon come across terms such as Brownian motion, random processes, measure, and Lebesgue integral. Based on the many years of experience we have gained in university teaching, we claim that some readers do not have sufficient knowledge in this field, unless they have studied mathematics. Therefore, they may not know what is meant by probability measures, Brownian motions, and similar terms.
Andreas Löffler, Lutz Kruschwitz

Open Access

2. Set Theory

Abstract
We will present the most important elements of set theory, because without appropriate knowledge one cannot acquire a sufficient understanding of Brownian motion. Set theory is also needed when it comes to the theory of random variables, probability theory, information economics, or game theory. Since set theory is not dealt with in sufficient detail in formal training of economists, we will discuss the required issues here.
Andreas Löffler, Lutz Kruschwitz

Open Access

3. Measures and Probabilities

Abstract
Continuous-time theory makes use of a sophisticated functional analytical apparatus. If you really want to understand what a Brownian motion is and how to use it, you have no choice but to first deal with measurement theory and general integration theory.
Andreas Löffler, Lutz Kruschwitz

Open Access

4. Random Variables

Abstract
Students of economics are confronted with random variables very early in their programs. They are confronted with this term not only in statistics and econometrics but practically in all economic subdisciplines, in particular in microeconomics and finance. The meaning of a random variable, however, remains somewhat vague. It is usually considered sufficient if students understand it to be data whose actual value is not guaranteed. However, we will not remain on the surface but provide more fundamental insights of random variables. The reader will learn that random variables are functions with specific properties.
Andreas Löffler, Lutz Kruschwitz

Open Access

5. Expectation and Lebesgue Integral

Abstract
In the previous chapter we dealt with the concept of probability in the context of any event space Ω. We described how to proceed appropriately to define a probability as a measure of a set. Now we are focusing on the determination of expectations and variances.
Andreas Löffler, Lutz Kruschwitz

Open Access

6. Wiener’s Construction of the Brownian Motion

Abstract
In many economic textbooks which address the Brownian motion one finds representations resembling those of Fig. 6.1. Let us initially focus on the blue path, a function frequently used to illustrate a typical path of a Brownian motion. Almost everyone would accept that the price of a share could develop as shown. In particular, economists find such a representation plausible. However, such an interpretation is more likely to mislead rather than to contribute to the understanding of what the Brownian motion is all about. Even worse, they convey a misconception of the Brownian motion. Let us explain this phenomenon by looking at a coin toss.
Andreas Löffler, Lutz Kruschwitz

Open Access

7. Supplements

Abstract
Anyone writing a book will rarely follow a plan that was not revised several times during the process. This was definitely the case when this book was written. We have discussed many different versions before we arrived at the current format. In some of these versions mathematical terms like “convergence of functions” or “cardinality of sets” played an important role. At the end, we found a way to discuss the Brownian motion without using these terms explicitly. The obvious consequence could have been to simply drop this material.
Andreas Löffler, Lutz Kruschwitz

Backmatter

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