Probability for Finance

EKKEHARD KOPP
University of Hull, Hull, UK

JAN MALCZAK
AGH University of Science and Technology, Krak′ow, Poland

TOMASZ ZASTAWNIAK
University of York, York, UK


Preface page vii
1 Probability spaces 1
1.1 Discrete examples 1
1.2 Probability spaces 6
1.3 Lebesgue measure 11
1.4 Lebesgue integral 13
1.5 Lebesgue outer measure 33
2 Probability distributions and random variables 39
2.1 Probability distributions 39
2.2 Random variables 46
2.3 Expectation and variance 56
2.4 Moments and characteristic functions 62
3 Product measure and independence 66
3.1 Product measure 67
3.2 Joint distribution 73
3.3 Iterated integrals 75
3.4 Random vectors in Rn 81
3.5 Independence 83
3.6 Covariance 96
3.7 Proofs by means of d-systems 98
4 Conditional expectation 106
4.1 Binomial stock prices 106
4.2 Conditional expectation: discrete case 112
4.3 Conditional expectation: general case 119
4.4 The inner product space L2(P) 130
4.5 Existence of E(X | G) for integrable X 137
4.6 Proofs 142
5 Sequences of random variables 147
5.1 Sequences in L2(P) 147
5.2 Modes of convergence for random variables 156
5.3 Sequences of i.i.d. random variables 167
5.4 Convergence in distribution 170
5.5 Characteristic functions and inversion formula 174
5.6 Limit theorems for weak convergence 176
5.7 Central Limit Theorem 180
Index 187