Stochastic Differential Equations and Diffusion Processes
1st Edition
Published on 18. August 2011
480 pages
978-0-08-096012-8 (ISBN)
Description
Stochastic Differential Equations and Diffusion Processes
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Content
- Front Cover
- Stochastic Differential Equations and Diffusion Processes
- Copyright Page
- Contents
- Preface
- General Notation
- Chapter I. 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. Stochastic processes adapted to an increasing family of sub s-fields
- 6. Martingales
- 7. Brownian motions
- 8. Poisson random measure
- 9. Point processes and Poisson point processes
- Chapter II. Stochastic integrals and Itô's formula
- 1. Itô's definition of stochastic integrals
- 2. Stochastic integrals with respect to martingales
- 3. Stochastic integrals with respect to point processes
- 4. Semi-martingales
- 5. Itô's formula
- 6. Martingale characterization of Brownian motions and Poisson point processes
- 7. Representation theorem for semi-martingales
- Chapter III. 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
- 5. Exponential martingales
- Chapter IV. Stochastic differential equations
- 1. Definition of solutions
- 2. Existence theorem
- 3. Uniqueness theorem
- 4. Solution by transformation of drift and by 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 V. 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. Malliavin's stochastic calculus of variation for Wiener functionals
- 8. The case of stochastic differential equations and hypoellipticity problem of heat equations
- Chapter VI. Theorems on comparison and approximation and their applications
- 1. A comparison theorem for one-dimensional Itô 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 lime 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
