Stochastic Differential Equations and Diffusion Processes

Description

Being a systematic treatment of the modern theory of stochastic integrals and stochastic differential equations, the theory is developed within the martingale framework, which was developed by J.L. Doob and which plays an indispensable role in the modern theory of stochastic analysis.

A considerable number of corrections and improvements have been made for the second edition of this classic work. In particular, major and substantial changes are in Chapter III and Chapter V where the sections treating excursions of Brownian Motion and the Malliavin Calculus have been expanded and refined. Sections discussing complex (conformal) martingales and Kahler diffusions have been added.

Front Cover 2
Stochastic Differential Equations and Diffusion Processes 5
Copyright Page 6
Dedication 7
Preface to the Second Edition 9
Preface 11
Table of Contents 15
General Notation 17
CHAPTER I. Preliminaries 19
- 1. Basic notions and notations 19
- 2. Probability measures on a metric space 20
- 3. Expectations, conditional expectations and regular conditional probabilities 29
- 4. Continuous stochastic processes 34
- 5. Stochastic processes adapted to an increasing family of sub σ -fields 38
- 6. Martingales 43

Front Cover 2
Stochastic Differential Equations and Diffusion Processes 5
Copyright Page 6
Dedication 7
Preface to the Second Edition 9
Preface 11
Table of Contents 15
General Notation 17
CHAPTER I. Preliminaries 19
- 1. Basic notions and notations 19
- 2. Probability measures on a metric space 20
- 3. Expectations, conditional expectations and regular conditional probabilities 29
- 4. Continuous stochastic processes 34
- 5. Stochastic processes adapted to an increasing family of sub σ -fields 38
- 6. Martingales 43
- 7. Brownian motions 58
- 8. Poisson random measure 60
- 9. Point processes and Poisson point processes 61
CHAPTER II. Stochastic Integrals and Ito's Formula 63
- 1. Itô's definition of stochastic integrals 63
- 2, Stochastic integrals with respect to martingales 71
- 3. Stochastic integrals with respect to point processes 77
- 4. Semi-martingales 81
- 5. Itô's formula 84
- 6. Martingale characterization of Brownian motions and Poisson point processes 91
- 7. Representation theorem for semi-martingales 102
CHAPTER III. Stochastic Calculus 115
- 1. The space of stochastic differentials 115
- 2. Stochastic differential equations with respect to quasimartingales 121
- 3. Moment inequalities for martingales 128
- 4. Some applications of stochastic calculus to Brownian motions 131
- 5. Exponential martingales 167
- 6. Conformal martingales 173
CHAPTER IV. Stochastic Differential Equations 177
- 1. Definition of solutions 177
- 2. Existence theorem 185
- 3. Uniqueness theorem 196
- 4. Solution by transformation of drift and by time change 208
- 5. Diffusion processes 220
- 6. Diffusion processes generated by differential operators and stochastic differential equations 230
- 7. Stochastic differential equations with boundary conditions 235
- 8. Examples 250
- 9. Stochastic differential equations with respect to Poisson point processes 262
CHAPTER V. Diffusion Processes on Manifolds 265
- 1. Stochastic differential equations on manifolds 265
- 2. Flow of diffeomorphisms 272
- 3. Heat equation on a manifold 287
- 4. Non-degenerate diffusions on a manifold and their horizontal lifts 293
- 5. Stochastic parallel displacement and heat equation for tensor fields 315
- 6. The case with boundary conditions 326
- 7. Kahler diffusions 359
- 8. Malliavin's stochastic calculus of variation for Wiener functionals 367
- 9. Pull-back of Schwartz distributions under Wiener mappings and the regularity of induced measures (probability laws) 393
- 10. The case of stochastic differential equations: Applications to heat kernels 409
CHAPTER VI. Theorems on Comparison and Approximation and their Applications 455
- 1. A comparison theorem for one-dimensional ltd processes 455
- 2. An application to an optimal control problem 459
- 3. Some results on one-dimensional diffusion processes 464
- 4. Comparison theorem for one-dimensional projection of diffusion processes 470
- 5. Applications to diffusions on Riemannian manifolds 478
- 6. Stochastic line integrals along the paths of diffusion processes 485
- 7. Approximation theorems for stochastic integrals and stochastic differential equations 498
- 8. The support of diffusion processes 535
- 9. Asymptotic evaluation of the diffusion measure for tubes around a smooth curve 550
Bibliography 559
Index 569

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