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.

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.

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.

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.

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.

Nach einer Grundlagenvorlesung über Wahrscheinlichkeitstheorie eignet sich das Puch perfekt zum Selbststudium, da sogar Lösungsansätze mit aufgezeigt werden. Das Buch hat 5 Sterne verdient und kann als Standardwerk in diesem Bereich angesehen werden.

The coverage of topics for an applied focus, or as a taster for pursuits more pure in the world of SDEs, is excellent. If more (e.g. analysis level) rigour is needed, one can supplement study with Durrett, or Rogers & Williams. What this book does well (which I originally hadn't appreciated) is to string together the topics well, and pitch them at a level which is not too abstruse. This is no small feat! Also, the exercises cover quite a lot of ground. This could be a good or bad thing depending on your predisposition.

testo molto didattico: l'autore non trascura il rigore matematico senza appesantire il testo di dimostrazioni (si rimanda alle referenze o all'appendice). Il filo del discorso è fluido e continuo, in modo da non perdere mai il fine ultimo, caratteristica che fa del libro anche un eccellente testo per autodidatti. La ricchezza di esercizi, svolti e non, alla fine di ogni capitolo aiuta il lettore ad una più profonda comprensione degli argomenti. L'approccio alla materia è generale ma non mancano negli ultimi capitoli applicazioni, soprattutto nel campo principe della matematica finanziaria. Consigliato soprattutto a tutti coloro che muovono i primi passi nel campo del calcolo stocastico e in genere a chi vuole con sé una pietra miliare dell'argomento.

This is a very good introduction on the subject, with interesting examples and a well-constructed theory. A must in the basic of stochastic calculus.

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