STOCHASTIC DIFFERENTIAL EQUATIONS AND DIFFUSION PROCESSES, 2ND ED. North-Holland Mathematical Library 24 Title Page Copyright Page Dedication Preface to the Second Edition Preface Contents General Notation Chapter 1. Preliminaries 1. Basic notions and notations 2. Probability measures on a metric space 3. Expectations, conditional expectations and regular conditional probabilities 4. Continuous stochastic processes 5. 6. Martingales 7. Brownian motions 8. Poisson random measure 9. Point processes and Poisson point processes Chapter 2. Stochastic Integrals and It?s Formula 1. It?s definition of stochastic integrals 2. Stochastic integrals with respect to martingales 3. Stochastic integrals with respect to point processes 4. Semi-martingales 5. It?s formula 6. Martingale characterization of Brownian motions and Poisson point processes 7. Representation theorem for semi-martingales Chapter 3. Stochastic Calculus 1. The space of stochastic differentials 2. Stochastic differential equations with respect to quasimartingales 3. Moment inequalities for martingales 4. Some applications of stochastic calculus to Brownian motions 4.1. Brownian local time 4.2. Reflecting Brownian motion and the Skorohod equation 4.3. Excursions of Brownian motion 4.4. Some limit theorems for occupation times of Brownian motion 5. Exponential martingales 6. Conformal martingales Chapter 4. Stochastic Differential Equations 1. Definition of solutions 2. Existence theorem 3. Uniqueness theorem 4. Solution by transformation of drift and by time change 4.1. The transformation of drift 4.2. Time change 5. Diffusion processes 6. Diffusion processes generated by differential operators and stochastic differential equations 7. Stochastic differential equations with boundary conditions 8. Examples 9. Stochastic differential equations with respect to Poisson point processes Chapter 5. Diffusion Processes on Manifolds 1. Stochastic differential equations on manifolds 2. Flow of diffeomorphisms 3. Heat equation on a manifold 4. Non-degenerate diffusions on a manifold and their horizontal lifts 5. Stochastic parallel displacement and heat equation for tensor fields 6. The case with boundary conditions 7. KÒler diffusions 8. Malliavin's stochastic calculus of variation for Wiener functionals 9. Pull-back of Schwartz distributions under Wiener mappings and the regularity of induced measures (probability laws) 10. The case of stochastic differential equations: Applications to heat kernels Chapter 6. Theorems on Comparison and Approximation and their Applications 1. A comparison theorem for one-dimensional It?processes 2. An application to an optimal control problem 3. Some results on one-dimensional diffusion processes 4. Comparison theorem for one-dimensional projection of diffusion processes 5. Applications to diffusions on Riemannian manifolds 6. Stochastic line integrals along the paths of diffusion processes 7. Approximation theorems for stochastic integrals and stochastic differential equations 8. The support of diffusion processes 9. Asymptotic evaluation of the diffusion measure for tubes around a smooth curve Bibliography Index 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. By Nobuyuki Ikeda And Shinzo Watanabe. Includes Index. Bibliography: P. 541-550.
