Stochastic-calculus

This book might just be the first ideal reading that students having to struggle with more advanced texts should do. The level is truly elementary and can be understood with the minimal 1-year college background, which is quite a feat compared with other books with similar claims. Examples are abundant and complement the pedagogically brilliant exposition by making everything intuititive. The style and level is reminiscent of Sheldon Ross' classics in probability and stochastic processes. However, even though advanced topics such as Paley-Wiener representation, Stratonovitch integral and numerical integration schemes are (alas too briefly) covered, the section about finance is disappointing and way too short. In summary an excellent book but look somewhere else for finance applications. And beg Prof. Mikosch for doubling the number of pages in the next edition...

Not having a strong theoretical mathematics background hindered my ability to read advanced stochastic finance. I found most "introduction to financial mathematics" for derivatives either too elementary or too advanced (i.e. unreadable). Mikosch has done an outstanding job of explaining key concepts of stochastic calculus, without losing a mathematically unsophisticated reader. After reading this book, one should feel comfortable reading more advanced texts on derivatives, which are usally full of mathematical jargon. I think, it's more suitable for readers with economics or engineering backgrounds who want to further explore the world of financial derivatives. If you have strong background in Analysis and Measure Theory, you might find this book too slow and not detailed enough (but then you are not the intended audience). Also, the book in itself is just an entry point into stochastic calculus and you'll need more advanced/theoretical texts on derivatives after. In my opinion, the book is not suitable for people who just want preliminary knowledge of derivatives; they should look for broader finance books, which usually have a few chapters on derivatives.

This is the most readable, and technical, introduction to stochastic calculus that I'm aware of. It doesn't really matter that finance is far from you. You can learn all the finance that you want, at an equally pedagogical level from Bjork's book, to give an example, as soon as you finish this. The only thing I miss in this book are equally pedagogical exercises + worked solutions. I hope that the next edition will be twice as large.

The parts of this book I've read have been clear and accessible for someone with an undergraduate degree in mathematics and some knowledge of stochastic processes. It doesn't needlessly multiply the jargon like some books, and it focuses mainly on the one-dimensional case so that the intuition isn't constantly obscured by matrix notation. Many subjects also have chatty introductions that offer intuition and a bit of relief from the hard work involved in learning this subject.

Some books are meant to teach, and to elucidate new material; this book is not one of them. It seems the purpose of this book was rather to record for prosperity all theorems related to Stochastic Calculus. Instead of developing any intuition on the subject, the author seems to think the purpose of writing is to use the most elegant proofs with the most modern of mathematical jargon. In short, the book consists of stated lemmas and theorems with terse, undeveloped proofs. This book will not teach you anything.

This is a math book first that happens to use math on aspects of finance as a topic. It is not a course for to teach people finance. It is a very elegant and sophisticated book for those who are very well versed in the necessary mathematics in stochastic calculus and in particular Martingale theory to show them how these tools can be applied to problems in finance.

If you have the math background and are interested in this topic you will get a lot from this book. If you don't have the math, don't bother. This book will be opaque.

I completely disagree with Student.

This book is indeed meant for learning. Just do not take it as your first entry into Stochastic Calculus. Take it as a second reading. It is complete, thorough and well, very well written.

It will teach you. A lot. All theorems are cross-referenced, so you will not have any "it is obvious that" etc. Theorems are proved, over and over again, until they hammer themselves in your head.

It is a fine achievement, if you want something quick and dirty read something else.

I recommend this book to anyone who wants to develop a deep understanding of Browninan Motion and Stochastic Calculus. However, the level of detail and rigor can obscure the main ideas, and so it is a very difficult introductory text for readers without a strong background in probability theory and continuous Markov processes. As a teaching assistant in a Mathematical Finance Masters program, I recommend that my students read Oskendal's Stochastic Differential Equations first, which gives an excellent introduction to the material without sacrificing rigor.

The theory of Brownian motion is ubiquitous in physics and mathematics, and has recently become very important in mathematical finance and network modeling. The observation of the irregular movement of pollen suspended in water by Robert Brown in 1828 led Albert Einstein to formulate a theory for Brownian motion. In this book the authors outline rigorously the theory of Browian motion. Their logic is impeccable, and the content is fascinating reading, even to those very experienced in the subject.

The authors begin in chapter 1 with the task of defining martingales and filtrations, with the notion of a stochastic process being adapted to a filtration taking on particular importance. They omit the proof that a process is progressively measurable if and only if it is measurable and adapted, because of the difficulty of the proof, but give a reference where the proof can be found. Continuous-time martingales are defined, with (compensated) Poisson processes given as an example. The Doob-Meyer decomposition and square-integrable martingales are discussed, and the chapter if full of exercises, with solutions provided to some of these at the end of the chapter. Brownian motion is formally defined in the next chapter, with its existence proven using Wiener measure on the space of continuous functions on the positive half line. The discussion in this chapter has to rank as one of the best in print, due to the meticulous and precise manner in which the material is presented. The Markov property of Brownian motion is proven, along with a good presentation of the Levi modulus of continuity. Readers working in constructive quantum field theory will see their usual construction of Wiener measure in the second exercise of the chapter. Those working in that area are used to seeing (conditional) Wiener measure defined on a collection of cylinder sets, which is then extended to the Borel subsets . Such a construction is done in this book, but the approach is somewhat different than what physicists normally see in quantum field theory.

The theory of stochastic integration is presented in Chapter 3, and it is superbly written. The authors are careful to distinguish the theory of integration for stochastic processes from the ordinary one with emphasis on the actual computation of stochastic integrals. The reader is first asked to explore the Stratonovitch and Ito integrals in an exercise., and then a thorough treatment is given by the authors later in the chapter. The authors point out the differences between the Ito and Stratonovich integrals, with the latter being defined for a smaller class of functions than the former. The important Ito rule for changing variables is discussed, and then used to give the Kunita-Watanabe martingale characterization of Brownian motion. Physicists involved in constructive quantum field theory will appreciate the discussion of the Trotter existence theorem in this chapter.

The connection of Brownian motion with partial differential equations, so familiar to physicists via the heat equation, is the subject of the next chapter. These equations give the transition probabilities of the stochastic process, and are studied here first in the context of harmonic analysis, namely the classical Dirichlet problem. This is followed by a beautiful treatment of the one-dimensional heat equation and the Feynman-Kac formulas. Those readers working in constructive quantum field theory will see the Green's function lurking in the background.

The very important topic of stochastic differential equations is outlined in chapter 5, with emphasis placed on the study of diffusive processes. The solutions of these equations have an immense literature, and the authors do not of course overview all of it, but do give a useful introduction. Both strong and weak solutions are discussed, with the Girsanov and Yamada-Watanabe techniques used throughout. Explicit solutions are given for linear stochastic differential equations, such as the Ornstein-Uhlenbeck process governing the Brownian motion of a particle with friction. Financial engineers will appreciate the discussion of the applications of this formalism to option pricing and the Merton consumption theory in this chapter. Options pricing is cast in martingale terms, and then the usual Black-Scholes equation is derived from this. The notorious Hamilton-Jacobi-Bellman equation is discussed in the consumption/investment problem, and the authors show how to employ techniques for solving this problem instead of solving this difficult nonlinear equation. The authors give a hint of the important Malliavin calculus in the Appendix and give references for the reader.

The last chapter of the book is more specialized than the rest and deals with the Levy theory of Brownian local time. This theory does have a connection with the theory of jump processes, which are currently very important in financial and network modeling. The authors do a fine job of explaining how Poisson random measures permit the event bookkeeping in these jump processes. Their discussion is applied to the computing of the transition probabilities for a Brownian motion with two-valued drift.

Before young talents try to prove themselves in this area they must know what is known... Karatzas and Shreve offered an intensive, rigorous and covering text on stochastic processes. Especially needed for quants working in the hottest applications, economics and finance, where demand is rapidly growing, but the qualifications should essentially be based on "Brownian Motion..." Warning: the book is for advanced readers; be ready for some overwhelming notations. Harder-than-usual efforts are required, by they will ultimately pay off.