Stochastic Differential Equations and Diffusion Processes
| Date: 15 April 2011, 23:54
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Contents
Chapter I. Preliminaries 1
1. Basic notions and notations 1
2. Probability measures on a metric space 2
3. Expectations, conditional expectations and regular conditional probabilities 11
4. Continuous stochastic processes 16
5. Stochastic processes adapted to an increasing family of sub a-fields 20
6. Martingales 25
7. Brownian motions 40
8. Poisson random measures 42
9. Point processes and Poisson point processes 43
Chapter II. Stochastic integrals and Ito's formula 45
1. Ito's definition of stochastic integrals 45
2. Stochastic integrals with respect to martingales 53
3. Stochastic integrals with respect to point processes, 59
4. Semi-martingales 63
5. Ito's formula 66
6. Martingale characterization of Brownian motions and Poisson point processes 73
7. Representation theorem for semi-martingales 84
Chapter III. Stochastic ca1culus 97
1. The space of stochastic differentials 97
2. Stochastic differential equations with respect to quasimartingales 103
3. Moment inequalities for martingales 110
4. Some applications of stochastic calculus to Brownian motions 113
4.1. Brownian local time 113
4.2. Reflecting Brownian motion and the Skorohod equation 119
4.3. Excursions of Brownian motion 123
4.4. Some limit theorems for occupation times of Brownian motion 136
5. Exponential martingales 140
Chapter IV. Stochastic differential equations 145
1. Definition of solutions 145
2. Existence theorem 153
3. Uniqueness theorem 164
4. Solution by transformation of drift and by time change 176
5. Diffusion processes 188
6. Diffusion processes generated by differential operators and stochastic differential equations 198
7. Stochastic differential equations with boundary conditions 203
8. Examples 218
9. Stochastic differential equations with respect to Poisson point processes 230
Chapter V. Diffusion processes on manifolds 233
1. Stochastic differential equations on manifolds 233
2. Flow of diffeomorphisms 239
3. Heat equation on a manifold 254
4. Non-degenerate diffusions on a manifold and their horizontal lifts 260
5. Stochastic parallel displacement and heat equation for tensor fields 282
6. The case with boundary conditions 289
7. Malliavin's stochastic calculus of variation for Wiener functionals 322
8. The case of stochastic differential equations and hypoellipticity problem of heat equations 334
Chapter VI. Theorems on comparison and approximation and their applications 352
1. A comparison theorem for one-dimensional Ito processes 352
2. An application to an optimal control problem 356
3. Some results on one-dimensional diffusion processes 361
4. Comparison theorem for one-dimensional projection of diffusion processes 367
5. Applications to diffusions on Riemannian manifolds 375
6. Stochastic line integrals along the paths of diffusion processes 382
7. Approximation theorems for stochastic integrals and stochastic differential equations 392
8. The support of diffusion processes 429
9. Asymptotic evaluation of the diffusion measure for tubes around a smooth curve 444
Bibliography 453
Index 461
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