Стохастические дифференциальные уравнения : введение в теорию и приложения

The main new feature of the fifth edition is the addition of a new chapter, Chapter 12, on applications to mathematical finance. I found it natural to include this material as another major application of stochastic analysis, in view of the amazing development in this field during the last 10-20 years. Moreover, the close contact between the theoretical achievements and the applications in this area is striking. For example, today very few firms (if any) trade with options without consulting the Black & Scholes formula! The first 11 chapters of the book are not much changed from the previous edition, but I have continued my efforts to improve the presentation through­ out and correct errors and misprints. Some new exercises have been added. Moreover, to facilitate the use of the book each chapter has been divided into subsections. If one doesn't want (or doesn't have time) to cover all the chapters, then one can compose a course by choosing subsections from the chapters. The chart below indicates what material depends on which sections. Chapter 6 Chapter IO Chapter 12 For example, to cover the first two sections of the new chapter 12 it is recom­ mended that one (at least) covers Chapters 1-5, Chapter 7 and Section 8.6. VIII Chapter 10, and hence Section 9.1, are necessary additional background for Section 12.3, in particular for the subsection on American options. This Book Gives An Introduction To The Basic Theory Of Stochastic Calculus And Its Applications. Examples Are Given Throughout The Text, In Order To Motivate And Illustrate The Theory And Show Its Importance For Many Applications In E.g. Economics, Biology And Physics. The Basic Idea Of The Presentation Is To Start From Some Basic Results (without Proofs) Of The Easier Cases And Develop The Theory From There, And To Concentrate On The Proofs Of The Easier Case (which Nevertheless Are Often Sufficiently General For Many Purposes) In Order To Be Able To Reach Quickly The Parts Of The Theory Which Is Most Important For The Applications. The New Feature Of This 5th Edition Is An Extra Chapter On Applications To Mathematical Finance. Introduction -- Some Mathematical Preliminaries -- Ito Integrals -- Ito Processes And The Ito Formula -- Stochastic Differential Equations -- The Filtering Problem -- Diffusions: Basic Problems -- Other Topics In Diffusion Theory -- Applications To Boundary Value Problems -- Applications To Optimal Stopping -- Application To Stochastic Control -- Application To Mathematical Finance -- Appendix A: Normal Random Variables -- Appendix B: Conditional Expectations -- Appendix C: Uniform Integrability And Martingale Convergence -- Solutions And Additional Hints To Some Of The Exercises -- Bibliography -- List Of Frequently Used Notation And Symbols -- Index. By Bernt Øksendal. The author, a lucid mind with a fine pedagogical instinct, has written a splendid text. He starts out by stating six problems in the introduction in which stochastic differential equations play an essential role in the solution. Then, while developing stochastic calculus, he frequently returns to these problems and variants thereof and to many other problems to show how the theory works and to motivate the next step in the theoretical development. Needless to say, he restricts himself to stochastic integration with respect to Brownian motion. He is not hesitant to give some basic results without proof in order to leave room for "some more basic applications..." . The book can be an ideal text for a graduate course, but it is also recommended to analysts (in particular, those working in differential equations and deterministic dynamical systems and control) who wish to learn quickly what stochastic differential equations are all about. The new edition of this bestselling book introduces the basic theory of stochastic calculus and its applications. Examples are given throughout to illustrate the theory and to show its importance for many applications that arise in areas such as economics, finance, physics, and biology. A new chapter on mathematical finance is included This text gives an introduction to the basic theory of stochastic calculus and its applications. Examples are given throughout the text, in order to motivate and illustrate the theory and show its importance for many applications, for example, economics, biology and physics.