Order from a Major Online Bookstore
Ultimate Book Price Comparison Engine
Save Your Time And Money
| AnyBook4Less.com | Order from a Major Online Bookstore |
|
| Home | Store List | FAQ | Contact Us | | ||
| Ultimate Book Price Comparison Engine Save Your Time And Money |
||
 |
Title: Brownian Motion and Stochastic Calculus (Graduate Texts in Mathematics, Vol 113) by Ioannis Karatzas, Steven E. Shreve, Steven E. Shreve ISBN: 0-387-97655-8 Publisher: Springer Verlag Pub. Date: 20 June, 1997 Format: Paperback Volumes: 1 List Price(USD): $59.95 |
Average Customer Rating: 4.67 (3 reviews)
Rating: 5
Summary: The best introduction
Comment: 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.
Rating: 4
Summary: A rigorous but difficult presentation of SDEs
Comment: 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.
Rating: 5
Summary: a must reading for quants working in economics and finance
Comment: 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.
 |
Title: Stochastic Differential Equations: An Introduction With Applications (Universitext) by B. K. Ksendal, Bernt Oksendal ISBN: 3540047581 Publisher: Springer Verlag Pub. Date: December, 2003 List Price(USD): $44.95 |
 |
Title: Methods of Mathematical Finance by Ioannis Karatzas, Steven E. Shreve ISBN: 0387948392 Publisher: Springer Verlag Pub. Date: 13 August, 1998 List Price(USD): $79.95 |
 |
Title: Dynamic Asset Pricing Theory, Third Edition. by Darrell Duffie ISBN: 069109022X Publisher: Princeton Univ Pr Pub. Date: 01 November, 2001 List Price(USD): $80.00 |
 |
Title: Stochastic Calculus and Financial Applications by J. Michael Steele ISBN: 0387950168 Publisher: Springer Verlag Pub. Date: 12 October, 2000 List Price(USD): $79.95 |
 |
Title: Continuous-Time Finance by Robert C. Merton ISBN: 0631185089 Publisher: Blackwell Publishers Pub. Date: June, 1992 List Price(USD): $51.95 |
Thank you for visiting www.AnyBook4Less.com and enjoy your savings!
Copyright� 2001-2021 Send your comments
