The book is meant to help the research student reach the stage where he or she can begin both to think up and tackle new problems and to read the up-to-date literature across a wide spectrum; and to persuade him or her that it is worth the effort.
We can say that we ourselves find the subject suffciently good fun to have enjoyed the task of writing. (We even had some amusement from typing the manuscript ourselves with the very basic non-mathematical word-processor VIEW on the BBC micro. Occasionally, we got into trouble when trying to use global editing to substitute the most commonly occurring phrases for shorthand versions of our own devising. But, in the main, we were very satito's formulaied!)
Chapter 4 Introduction To Ito Calculus
1. Some Motivating Remarks
2. Some Fundamental Ideas: Previsible Processes, Localization
3. The Elementary Theory of Finite-Variation
4. Stochastic Integrals: The L Theory
5. Stochastic Integrals With Respect To Continuous Semimartingales
6. Applications Of Ito's Formula
Chapter 5 Stochastic Differential Equations and Diffusions
1. introduction
2. Pathwise Uniqueness, Strong SDEs, And Flows
3. Weak Solutions, Uniqueness In Law
4. Martingale Problems, Markov Property
5. Overture To Stochastic Differential Geometry
6. One-Dimensional SDEs
7. One-Dimensional Diffusions
Chapter 6 The General Theory
1. Orientation
2. Debut And Section Theorems
3. Optional Projections And Filtering
4. Characterizing Previsible Times
5. Dual Previsible Projections
6. The Meyer Decomposition Theorem
7. Stochastic Integration: The General Case
8. Ito Excursion Theory