金融中的概率与随机过程导论(英文版——FABIO TROJANI)
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Contents
1 Introduction to Probability Theory
4
1.1 The Binomial Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.1.1 The Risky Asset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3.3 Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.3.4 Expected Value and Lebesgue Integral . . . . . . . . . . . . . . . . . . . . . 25
2 Conditional Expectations and Martingales
33
2.1 The Binomial Model Once More . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
Βιβλιοθήκη Baidu
2.2 Sub Sigma Algebras and (Partial) Information . . . . . . . . . . . . . . . . . . . . 34
∗Correspondence address: Fabio Trojani, Swiss Institute of Banking and Finance, University of St. Gallen, Rosenbergstr. 52, CH-9000 St. Gallen, e-mail: Fabio.Trojani@unisg.ch.
1.2.4 Expected Value of Random Variables Defined on Finite Measurable Spaces 15
1.2.5 Examples of Probability Spaces and Random Variables with Finite Sample
1.1.4 Some Basic Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.1.5 Pricing Derivatives: a first Example . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Finite Probability Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2.1 Measurable Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.3 General Probability Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.2.2 Probability measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.2.3 Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
Introduction to Probability Theory and Stochastic Processes for Finance∗ Lecture Notes
Fabio Trojani Department of Economics, University of St. Gallen, Switzerland
1.1.2 The Riskless Asset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.1.3 A Basic No Arbitrage Condition . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3.5 Some Further Examples of Probability Spaces with uncountable Sample Spaces 28
1.4 Stochastic Independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
1
2.3 Conditional Expectations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.3.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.3.2 Definition and Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
1.3.1 Some First Examples of Probability Spaces with non finite Sample Spaces . 18
1.3.2 Continuity Properties of Probability Measures . . . . . . . . . . . . . . . . 20