Description
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. Andreas Lffler received his postdoctoral qualification (habilitation) in Mathematics and Economics from the University of Leipzig and Free University Berlin, Germany, and has been a Professor of Banking and Finance at the Department of Finance, Accounting and Taxation of the Free University of Berlin since 2012. Lutz Kruschwitz is a Professor Emeritus of Banking and Finance at the Free University of Berlin, Germany. 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Stochastics in finance theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Precision and intuition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Purpose of the book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2 Set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.1 Notation and set operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.2 Events and sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3 Measures and probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.1 Basic problem of measurement theory . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.2 _-algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.3 Examples and interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.4 Further examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.5 Definition of a measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.6 Stieltjes measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.7 Dirac measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.8 Null sets and the almost-everywhere property . . . . . . . . . . . . . . . . . . . 48 4 Random variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 4.1 Random variables as functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 4.2 Random variables as measurable functions . . . . . . . . . . . . . . . . . . . . . 54 4.3 Distribution functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 5 Expectation and Lebesgue integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 5.1 Definition of expectation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 5.2 Riemann integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 5.3 Lebesgue integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 5.4 Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 5.5 Conditional expectation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 6 Wiener’s construction of the Brownian motion . . . . . . . . . . . . . . . . . . . . 77 6.1 Preliminary remark: the space of all paths . . . . . . . . . . . . . . . . . . . . . . 77 6.2 Wiener measure on C0;1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 6.3 Two definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 6.4 Neglected properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 7 Supplements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 7.1 Cardinality of sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 7.2 Continuity and differentiability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 7.3 Convergence terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 7.4 Once again: conditional expectations . . . . . . . . . . . . . . . . . . . . . . . . . . 103 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
