Tài liệu Steven Shreve: Stochastic Calculus and Finance

Steven Shreve: Stochastic Calculus and Finance


Contents


1 Introduction to Probability Theory 11


1.1 The Binomial Asset Pricing Model . . . . . . . . . . . . . . . . . . . . . . . . 11


1.2 Finite Probability Spaces . . . . . . . . . . . . . . . . . . . . . . . . . 16


1.3 Lebesgue Measure and the Lebesgue Integral . . . . . . . . . . . . . . . . 22


1.4 General Probability Spaces . . . . . . . . . . . . . . . . . . . . . . . . 30


1.5 Independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40


1.5.1 Independence of sets . . . . . . . . . . . . . . . . . . . . . . . 40


1.5.2 Independence of  -algebras . . . . . . . . . . . . . . . . . . . . . 41


1.5.3 Independence of random variables . . . . . . . . . . . . . . . . . . . . 42


1.5.4 Correlation and independence . . . . . . . . . . . . . . . . . . . . 44


1.5.5 Independence and conditional expectation. . . . . . . . . . . . . . 45


1.5.6 Law of Large Numbers . . . . . . . . . . . . . . . . . . . . . . . . 46


1.5.7 Central Limit Theorem . . . . . . . . . . . . . . . . . . . . . . . . 47


2 Conditional Expectation 49


2.1 A Binomial Model for Stock Price Dynamics . . . . . . . . . . . . . . . . 49


2.2 Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50


2.3 Conditional Expectation . . . . . . . . . . . . . . . . . . . . . . . . . 52


2.3.1 An example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52


2.3.2 Definition of Conditional Expectation . . . . . . . . . . . . . . . . 53


2.3.3 Further discussion of Partial Averaging . . . . . . . . . . . . . . . 54


2.3.4 Properties of Conditional Expectation . . . . . . . . . . . . . . . . 55


2.3.5 Examples from the Binomial Model . . . . . . . . . . . . . . . . . 57


2.4 Martingales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58


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