Stochastic Differential Equations: An Introduction with Applications (Universitext) (Anglais) Broché – 22 septembre 2010
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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. Lire la première page
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Another virtue of this book is the plenty (easy) exercise problems. Working through them is perhaps the best way to learn stochastic calculus.
There are a number of complaints to be made about this book. Most importantly is that in his attempt at simplification, Oksendal frequently chooses shedding (important) details over properly motivating a new concept. I found this particularly true in his exposition of generators. The book is poorly also organized: a number of topics are arbitrarily split into different chapters, important ideas hide inside of examples, etc.
While this is not my favorite book by any means, there is currently no replacement for it. Jumping directly into a book like Karatzas&Shreeve can be daunting. I would recommend getting a used copy. Also, previous editions seem to be very nearly identical to the current edition.
I also recommend checking out Rogers&Williams "Diffusions, Markov Process, and Martingales" Vols I&II.
I would much rather recommend Shreve's Stochastic Calculus for Finance II. Though longer, it is much more well-motivated and gives you a more intuitive feel for the concepts as opposed to Oksendal's full-on theoreical treatment.
Stochastic Differential Equations is a branch of mathematics. This book is not just for financial derivatives analysis or modeling. Oksendal first introduces the subject by raising a few stochastic problems (population growth; electric charge in RLC circuit; filtering problems, Dirichlet problems; asset management; optimal portfolio and options pricing) in the first chapter. The subsequent chapters develop notions and techniques which are able to solve wide varieties of stochastic problems (not just those mentioned in the first chapters). The arrangement is impressive in particular for readers who have no previous knowledge about the subject. The readers at least know the target for developing the techniques and would not lose the way when manipulating tons of symbols. Hints and answers to selected problems are invaluable to students for self-study.
To achieve a sound background on stochastic equations is extremely important especially in quantitative finance. It is not an easy job however. QF students may consider going through this book before seriously take Shreve's books on Stochastic Calculus for Finance.
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