Stochastic Differential Equations and Diffusion Processes

Title Page......Page 2
Copyright Page......Page 3
Dedication......Page 4
Preface......Page 5
General Notation......Page 11
1. Basic notions and notations......Page 13
2. Probability measures on a metric space......Page 14
3. Expectations, conditional expectations and regular conditional probabilities......Page 23
4. Continuous stochastic processes......Page 28
5. Stochastic processes adapted to an increasing family of sub a-fields......Page 32
6. Martingales......Page 37
7. Brownian motions......Page 52
8. Poisson random measures......Page 54
9. Point processes and Poisson point processes......Page 55
1. Ito's definition of stochastic integrals......Page 57
2. Stochastic integrals with respect to martingales......Page 65
3. Stochastic integrals with respect to point processes,......Page 71
4. Semi-martingales......Page 75
5. Ito's formula......Page 78
6. Martingale characterization of Brownian motions and Poisson point processes......Page 85
7. Representation theorem for semi-martingales......Page 96
1. The space of stochastic differentials......Page 109
2. Stochastic differential equations with respect to quasimartingales......Page 115
3. Moment inequalities for martingales......Page 122
4.1. Brownian local time......Page 125
4.2. Reflecting Brownian motion and the Skorohod equation......Page 131
4.3. Excursions of Brownian motion......Page 135
4.4. Some limit theorems for occupation times of Brownian motion......Page 148
5. Exponential martingales......Page 152
1. Definition of solutions......Page 157
2. Existence theorem......Page 165
3. Uniqueness theorem......Page 176
4. Solution by transformation of drift and by time change......Page 188
5. Diffusion processes......Page 200
6. Diffusion processes generated by differential operators and stochastic differential equations......Page 210
7. Stochastic differential equations with boundary conditions......Page 215
8. Examples......Page 230
9. Stochastic differential equations with respect to Poisson point processes......Page 242
1. Stochastic differential equations on manifolds......Page 245
2. Flow of diffeomorphisms......Page 251
3. Heat equation on a manifold......Page 266
4. Non-degenerate diffusions on a manifold and their horizontal lifts......Page 272
5. Stochastic parallel displacement and heat equation for tensor fields......Page 294
6. The case with boundary conditions......Page 301
7. Malliavin's stochastic calculus of variation for Wiener functionals......Page 334
8. The case of stochastic differential equations and hypoellipticity problem of heat equations......Page 346
1. A comparison theorem for one-dimensional Ito processes......Page 364
2. An application to an optimal control problem......Page 368
3. Some results on one-dimensional diffusion processes......Page 373
4. Comparison theorem for one-dimensional projection of diffusion processes......Page 379
5. Applications to diffusions on Riemannian manifolds......Page 387
6. Stochastic line integrals along the paths of diffusion processes......Page 394
7. Approximation theorems for stochastic integrals and stochastic differential equations......Page 404
8. The support of diffusion processes......Page 441
9. Asymptotic evaluation of the diffusion measure for tubes around a smooth curve......Page 456
Bibliography......Page 465
Index......Page 473