金融用概率
Probability for Finance
Patrick Roger
英文版
2010



Contents
Introduction 8
1. Probability spaces and random variables 10
1.1 Measurable spaces and probability measures 10
1.1.1 σ algebra (or tribe) on a set Ω 11
1.1.2 Sub-tribes of A 13
1.1.3 Probability measures 16
1.2 Conditional probability and Bayes theorem 18
1.2.1 Independant events and independant tribes 19
1.2.2 Conditional probability measures 21
1.2.3 Bayes theorem 24
1.3 Random variables and probability distributions 25
1.3.1 Random variables and generated tribes 25
1.3.2 Independant random variables 29
1.3.3 Probability distributions and cumulative distributions 30
1.3.4 Discrete and continuous random variables 34
1.3.5 Transformations of random variables 35
2. Moments of a random variable 37
2.1 Mathematical expectation 37
2.1.1 Expectations of discrete and continous random variables 39
2.1.2 Expectation: the general case 40
2.1.3 Illustration: Jensen’s inequality and Saint-Peterburg paradox 43
2.2 Variance and higher moments 46
2.2.1 Second-order moments 46
2.2.2 Skewness and kurtosis 48
2.3 The vector space of random variables 50
2.3.1 Almost surely equal random variables 51
2.3.2 The space L1 (Ω, A, P) 53
2.3.3 The space L2 (Ω, A, P) 54
2.3.4 Covariance and correlation 59
2.4 Equivalent probabilities and Radon-Nikodym derivatives 63
2.4.1 Intuition 63
2.4.2 Radon Nikodym derivatives 67
2.5 Random vectors 69
2.5.1 Definitions 69
2.5.2 Application to portfolio choice 71
3. Usual probability distributions in financial models 73
3.1 Discrete distributions 73
3.1.1 Bernoulli distribution 73
3.1.2 Binomial distribution 76
3.1.3 Poisson distribution 78
3.2 Continuous distributions 81
3.2.1 Uniform distribution 81
3.2.2 Gaussian (normal) distribution 82
3.2.3 Log-normal distribution 86
3.3 Some other useful distributions 91
3.3.1 The X 2 distribution 91
3.3.2 The Student-t distribution 92
3.3.3 The Fisher-Snedecor distribution 93
4. Conditional expectations and Limit theorems 94
4.1 Conditional expectations 94
4.1.1 Introductive example 94
4.1.2 Conditional distributions 96
4.1.3 Conditional expectation with respect to an event 97
4.1.4 Conditional expectation with respect to a random variable 98
4.1.5 Conditional expectation with respect to a substribe 100
4.2 Geometric interpretation in L2 (Ω, A, P) 101
4.2.1 Introductive example 101
4.2.2 Conditional expectation as a projection in L2 102
4.3 Properties of conditional expectations 104
4.3.1 The Gaussian vector case 105
4.4 The law of large numbers and the central limit theorem 108
4.4.1 Stochastic Covergences 108
.4.2 Law of large numbers 109
4.4.3 Central limit theorem 112
Bibliography 114