Stochastic Differential Equations: An Introduction with Applications (Universitext)

This book is offers an excellent introduction to SDE but limiting the text to integration w.r.t Brownian motion.The book is structured by first introducing 6 problems which are solved using the concepts and theory discussed in the chapters that follow. This is an excellent pedagogical tool, that is used to focus the mind on applications, in order to understand the abstract concepts discussed.The level of mathematics is moderate in difficulty with some proofs omitted (but with references included) for the sake of not veering away too far from the main concepts (and the need to introduce further preliminaries to understand the proof).There are also exercises included (with some solutions and hints) that allows the reader to solidify the understanding and applications.The follow-up text is commonly the Karatzas and Shreve book,though its level of difficulty is substantially higher than this text.

A classic. Written with an advanced reader in mind, this book covers most topics of stochastic calculus in great detail and with sufficient clarity. Worked examples are very helpful. Unless your (graduate) degree included coursework in stochastic calculus, it is not easy reading. Definitely read it with pen and paper, otherwise a lot of the material will not sink in.

From the cover, one can infer that this book means business. Some books still try to be artistic to attract audiences, whereas this book does away with a creative cover altogether. How often do you see that a book's cover contains five sample paths of a geometric Brownian Motion? Inside, Oksendal writes very clearly and uses the same format throughout. Although the topic is not the easiest to understand, you can acquire the skills that would allow you to gain sufficient knowledge of stochastic differential equations. He starts off with a good introduction and then moves on to the main topics. His applications to finance are also very useful for those in the field. A word of caution is that you would need a decent background in mathematics to read this book, but it is easier than Shreve or Karatzas and Shreve.

From what I've seen, this text assumes knowledge of measure theoretic probability. The author does not ever (to my knowledge) explicitly state what the prerequisites to the book are, but he does state that the book is based off of notes for a course in which familiarity with measure theory is assumed. Thus if you are not a graduate math student this text is going to be too much for you.I have learned the subject of stochastic calculus from Calin's great text, and thus this text has become much more understandable to me. I'd say that this could be a decent second book on the subject after reading Calin. Some of the technical stuff here might still fly over your head but I think there's a lot to be learned from this book.It is written in the way that a typical math book at this level is usually written. Very direct and straight to the point. There are many proofs and examples throughout the book. The exercises are in the back of each chapter and the author does include some solutions!If you've learned stochastic calculus somewhere else or have completed your graduate level courses then this book is great for you. It is introductory at the graduate level.As far as applications of SDEs to the sciences I think there are better places to start. This book feels like it is written for the mathematician, not the scientist and engineer. Try Solin and Sarkka instead (assumes knowledge of probability).Finally, if you want a graduate level text that covers probability then Evans is probably a great choice. If you want a comprehensive text then check out Baldi. If you're not mathematically inclined then Calin is your best bet.

Misleading title - NOT AN INTRODUCTION. There are much better places to start with stochastic integration.(EDIT) Even after I circled back years later to this book, it is practically useless as a refresher or reference. The material is poorly organized and presented in a stark and unmotivated manner. The fundamentally compelling properties of Brownian motion are glossed over. Many important proofs are abbreviated, omitted, or presented in the most terse and "clean" form possible, which is NOT IDEAL for introduction. This book straddles that useless middle zone between basic application and full technicality. It would be impossible to learn from this book alone without a very helpful and knowledgeable professor. For self-study it is a horrible choice. For finance practitioners seeking technical grounding for no-arbitrage pricing theory applications, Shreve I & II build up the theory from an actual introductory (discrete) perspective and constantly channel the material towards economic application, whereas this book lops on an almost useless exposition of financial pricing theory at the very end, presented in full "market-wide" generality and without data examples, graphs, etc. For researchers I recommend Williams and Rodgers I & II, which are very very dense but at least try to be exhaustive and provide a much more compelling view of the interrelated concepts in stochastic calculus. The first chapter alone of pt I is like a breath of fresh air compared to this sterile collection of unmotivated mathematical facts.

The title says it all. It is an excellent book for beginners to get in to stochastic calculus. A small suggestion that you revise your ODE before you tackle this book as it will ease the references the author likes to make to ODE.