1. Brownian Motion
1.1 Definition and Construction 1
1.2 Markov Property, Blumenthal's 0-1 Law 7
1.3 Stopping Times, Strong Markov Property 18
1.4 First Formulas 26 i . '
2. Stochastic Integration
2.1 Integrands: Predictable Processes 33 :
2.2 Integtators: Continuous LocalMartingales 37
2.3 Vaiiance and Covariance Processes 42
2.4 Integrat,ion w.r.t. Bounded Martingales 52
2.5 The Kunita-Watanabelnequality 59 :; ;
2.6 Integration w.r.t. Local Martingales 63
2.7 Chaoge ofVatiables,lto's Formula 68
2.8 Integration w.r.t. Semimartingales 70
2.9 Associative Law 74
2~.10 Functions of Several Semi.martingales 76
Chapter Summary 79
2.11 Meyer-Tan_aka Formula, Local Time 82
2.12 Girsanov's Formula 90
3. Brownian Motion, II
3.1 Recurrence and IYansience 95
3.2 0ccupation Times 100
3.3 Exit Times 105
3.4 Change ofTime, Levy's Theorem 111
3.5 Burkholder Davis Gundylnequalities 116
3.6 Martingales Adapted to Brownian Filtrations 119
4. Partial Drfferential Equations
A. Parabolic Equations
4.1 The:EIeat Equation 126
4.2 Thelnhomogeneous Equation 130
4.3 The Feynman-Kac Formula 137
B. Elliptic Equations
4.4 The Dirichlet Problem 143
4.5 Poisson's Equation 151
4.6 The Schrodinger E;quation_ 156
C. Applications to Brownian Motion
4.7 Exit Distributions for the Bail 164
4.8 0ccupation Times for the Ball 167
4.9 Laplace Transforms, Arcsine Law 170
5. Stochastic DiflFerential Equations
5.1 Examples 177
5.2 Ito's Approach 183
5.3 lExtension 190
5.4 Weak Solutions 196
5.5 Change ofMeasure 202
5.6 Change ofTime 207
6. One Dimensional Drffusions
6.1onstruction_ 211
6.2 Feller's Test 214
6.3 Recurrence and thansience 219
6.4 Green's Functions 222
6.5 Boundary Behavior 229
6.6 Applications to lIighet Dimensions 234
7. Diffusions as Markov Processes
7.1 Semigroups and Generators 245
7.2 Examples 250
7.3tansition Probabilities 255
7.4 lE[anis Chains 258
7.5 Convergerice Theorems 268
8. Weak Convergence
8.1 In Metric Spaces 271
8.2 Prokhorov's Theorems 276
8.3 The Space C 282
8.4 Skorohod's Existence Theorem for SDE 285
8.5 Donsker's Theorem 287
8.6 The Space D 293
8.7 Convergence to DHfusions 296
8.8 Examples 305
Solutions to Exercises 311
References 335
Index 339
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