Contents i x
Chapter 8 . Martingale s 119
§1. Conditiona l Expectation s 119
§2. Filtration s an d Semi-Martingale s 126
§3. Stoppin g Time s an d Optiona l Stoppin g 129
§4. Application s t o Rando m Walk s 131
§5. Inequalitie s an d Convergenc e 134
§6. Furthe r Application s 136
Problems 15
Notes 157
Chapter 9 . Brownia n Motio n 159
§1. Gaussia n Processe s 160
§2. Wiener' s Construction : Brownia n Motio n o n [ 0 ,1) 165
§3. Nowhere-Differentiabilit y 168
§4. Th e Brownia n Filtratio n an d Stoppin g Time s 170
§5. Th e Stron g Marko v Propert y 173
§6. Th e Reflectio n Principl e 175
Problems 176
Notes 180
Chapter 10. Terminus : Stochasti c Integratio n 181
§1. Th e Indefinit e It o Integra l 181
§2. Continuou s Martingale s i n L
2(P)
187
§3. Th e Definit e It o Integra l 189
§4. Quadrati c Variatio n 192
§5. Ito' s Formul a an d Tw o Application s 193
Problems 199
Notes 20
Appendix 20 3
§1. Hilber t Space s 20 3
§2. Fourie r Serie s 20 5
Bibliography 20 9
Index 217
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