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.

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.

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.

It's hard to find anything good to say about this textbook. There are more equations than words in this book. The book was created from Oksendal's lecture notes and I'd say that about sums up the amount of explanation you are going to get in this textbook. No explanations, no motivation, just equations and complicated proofs that assume a lot of prior PhD level mathematics like measure theory. Also relies on some graduate level analysis and touches a little on PDEs and boundary value problems. The Mathematical Preliminaries section of the text is supposed to prepare you for the math needed for the text, however, the preliminaries section has little to do the math actually used in later chapters with just 5 short pages on probability theory and Brownian motion. How can this then be considered an introduction? I found Oksendal's text nothing but confusing and filled me with anxiety as I tried to hopelessly wade through the math feeling lost, angry, dumb and depressed. I do not recommend. I have yet to find a clear exposition of this topic, but this is by far the worst of the bunch. I'd look at Klebnar or Shreve II, both of which can be a challenge, but at least you feel like you are making some progress along the way.

The book makes us understand the actual importance of the probability.Today the books about the stochastic equations have superated the interest of the traditional analysis.The author explicates with competence the definition of the martingale, filter or Markov chain. The applications are about the finance, the control theory, the problem of stopping.

Oksendal suffers from measurement theory minuatae in order to make this a rigourous text. Frustatingly the author has economised in proofs, leaving out the 'unnecessary' intermediate steps etc wasting a lot of your time to reconstruct. If you've never seen an SDE before, read Elementary Stochastic Equations by Miksovich before attempting this 'Introduction' - really an intermediate text. I really didn't like this book, more could be done to make it comprehensible with less reader effort.