The Brownian Motion : A Rigorous but Gentle Introduction for Economists. - Löffler, Andreas; Kruschwitz, Lutz. (Cham : Springer International Publishing AG, 2019)
Intro -- Preface -- Acknowledgments -- Contents -- 1 Introduction -- 1.1 Stochastics in Finance Theory -- 1.2 Precision and Intuition in the Valuation of Derivatives -- 1.3 Purpose of the Book -- 2 Set Theory -- 2.1 Notation and Set Operations -- 2.2 Events and Sets -- 3 Measures and Probabilities -- 3.1 Basic Problem of Measurement Theory -- 3.2 σ-Algebras and Their Formal Definition -- 3.3 Examples of Measurable Sets and Their Interpretation -- 3.4 Further Examples: Infinite Number of States and Times -- 3.5 Definition of a Measure -- 3.6 Stieltjes Measure -- 3.7 Dirac Measure -- 3.8 Null Sets and the Almost-Everywhere Property -- 4 Random Variables -- 4.1 Random Variables as Functions -- 4.2 Random Variables as Measurable Functions -- 4.3 Distribution Functions -- 5 Expectation and Lebesgue Integral -- 5.1 Definition of Expectation: A Problem -- 5.2 Riemann Integral -- 5.3 Lebesgue Integral -- 5.4 Result: Expectation and Variance as Lebesgue Integral -- 5.5 Conditional Expectation -- 6 Wiener's Construction of the Brownian Motion -- 6.1 Preliminary Remark: The Space of All Paths -- 6.2 Wiener Measure on the Space of Continuous Functions -- 6.3 Two Definitions of the Brownian Motion -- 6.4 Often Neglected Properties of the Brownian Motion -- 7 Supplements -- 7.1 Cardinality of Sets -- 7.2 Continuous and Almost Nowhere Differentiable Functions -- 7.3 Convergence Terms -- 7.4 Conditional Expectations Are Random Variables -- References -- Index.
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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2023. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.
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