Stochastic Differential Equations

These notes are based on a postgraduate course I gave on stochastic differential equations at Edinburgh University in the spring 1982. No previous knowledge about the subject was assumed, but the presen­ tation is based on some background in measure theory. There are several reasons why one should learn more about stochastic differential equations: They have a wide range of applica­ tions outside mathematics, there are many fruitful connections to other mathematical disciplines and the subject has a rapidly develop­ ing life of its own as a fascinating research field with many interesting unanswered questions. Unfortunately most of the literature about stochastic differential equations seems to place so much emphasis on rigor and complete­ ness that is scares many nonexperts away. These notes are an attempt to approach the subject from the nonexpert point of view: Not knowing anything (except rumours, maybe) about a subject to start with, what would I like to know first of all? My answer would be: 1) In what situations does the subject arise? 2) What are its essential features? 3) What are the applications and the connections to other fields? I would not be so interested in the proof of the most general case, but rather in an easier proof of a special case, which may give just as much of the basic idea in the argument. And I would be willing to believe some basic results without proof (at first stage, anyway) in order to have time for some more basic applications.

I. Introduction. - II. Some Mathematical Preliminaries. - III. Ito Integrals. - IV. Stochastic Integrals and the Ito Formula. - V. Stochastic Differential Equations. - VI. The Filtering Problem. - VII. Diffusions. - VIII. Applications to Partial Differential Equations. - IX. Application to Optimal Stopping. - X. Application to Stochastic Control. - Appendix A: Normal Random Variables. - Appendix B: Conditional Expectations. - List of Frequently Used Notation and Symbols.