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The Brownian Motion: A Rigorous But Gentle Introduction for Economists
Paperback
Series: Springer Texts in Business and Economics
ISBN13: 9783030201050
Publisher: Springer Nature
Published: Aug 14 2020
Pages: 125
Weight: 0.44
Height: 0.29 Width: 6.14 Depth: 9.21
Language: English
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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 726 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 . . . . . . . . . . . . . . . . . . . .
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