Oksendal Stochastic differential equations

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. For The 6th Edition The Author Has Added Further Exercises And, For The First Time, Solutions To Many Of The Exercises Are Provided. This Corrected 6th Printing Of The 6th Edition Contains Additional Corrections And Useful Improvements, Based In Part On Helpful Comments From The Readers. Introduction -- Some Mathematical Preliminaries -- Itô Integrals -- Itô Formula And The Martingale Representation Theorem -- Stochastic Differential Equations -- The Filtering Problem -- Diffusions: Basic Properties -- 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 -- Appendix D: An Approximation Result -- Solutions And Additional Hints To Some Of The Exercises -- References -- List Of Frequently Used Notation And Symbols -- Index. Bernt Øksendal. Includes Bibliographical References (p. [345]-351) And Index. 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. 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 in order to quickly progress to the parts of the theory that are most important for the applications. For the 6th edition the author has added further exercises and, for the first time, solutions to many of the exercises are provided. If we allow for some randomness in some of the coefficients of a differential equation we often obtain a more realistic mathematical model of the situation.