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注意事项：这不是Steven Shreve的书，请注意辨别
MAREK CAPIN′ SKI
AGH University of Science and Technology, Krak′ow, Poland
EKKEHARD KOPP
University of Hull, Hull, UK
JANUSZ TRAPLE
AGH University of Science and Technology, Krak′ow, Poland
Preface page vii
1 Discrete-time processes 1
1.1 General definitions 2
1.2 Martingales 6
1.3 The Doob decomposition 11
1.4 Stopping times 14
1.5 Doob’s inequalities and martingale convergence 22
1.6 Markov processes 28
1.7 Proofs 34
2 Wiener process 36
2.1 Scaled random walk 36
2.2 Definition of the Wiener process 40
2.3 A construction of the Wiener process 41
2.4 Elementary properties 46
2.5 Stochastic processes: basic definitions 49
2.6 Properties of paths 51
2.7 Martingale properties 55
2.8 Doob’s inequalities 59
2.9 Stopping times 61
2.10 Markov property 66
2.11 Proofs 71
3 Stochastic integrals 78
3.1 Motivation 78
3.2 Definition of the Itˆo integral 80
3.3 Properties 88
3.4 Itˆo processes 91
3.5 Proofs 99
4 Itˆo formula 109
4.1 A heuristic derivation 109
4.2 Functions of the Wiener process 112
4.3 Functions of Itˆo processes 117
4.4 Extension to general F 126
4.5 Localising stopping times 129
4.6 Extension of the stochastic integral 131
4.7 The Itˆo formula for general integrands 136
4.8 Local martingales 140
4.9 Applications of the Itˆo formula 144
4.10 Proofs 148
5 Stochastic differential equations 152
5.1 Examples 153
5.2 Existence and uniqueness of solutions 160
5.3 Markov property 169
5.4 Proofs 174
Index 176
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