Product Description
The numerical analysis of stochastic differential equations differs significantly from that of ordinary differential equations, due to the peculiarities of stochastic calculus. The book proposes to the reader whose background knowledge is limited to undergraduate level methods for engineering and physics, and easily accessible introductions to SDE and then applications as well as the numerical methods for dealing with them. To help the reader develop an intuitive understanding and hand-on numerical skills, numerous exercises including PC-exercises are included. ... Read more

Customer Reviews (2)

A reference book in the domain
Much literature is published on numerical methods for stochastic differential systems but most of it focuses on their use in pricing financial products. There is genuinely a lack of reference books that provide a stronger mathematical basis for the domain. Luckily, this is one of the few books that fill that gap. An excellent book, although the scope of numerical methods presented is limited.

Excellent
This book is one of the finest written on the subject and is suitable for readers in a wide variety of fields, including mathematical finance, random dynamical systems, constructive quantum field theory, and mathematical biology. It is certainly well-suited for classroom use, and it includes computer exercises what are definitely helpful for those who need to develop actual computer code to solve the relevant equations of interest. Since it emphasizes the numerical solution of stochastic differential equations, the authors do not give the details behind the theory, but references are given for the interested reader.

As preparation for the study of SDEs, the authors detail some preliminary background on probability, statistics, and stochastic processes in Part 1 of the book. Particularly well-written is the discussion on random number generators and efficient methods for generating random numbers, such as the Box-Muller and Polar Marsaglia methods. Both discrete and continuous Markov processes are discussed, and the authors review the connection between Weiner processes (Brownian motion for the physicist reader) and white noise. The measure-theory foundations of the subject are outlined briefly for the interested reader.

Part 2 begins naturally with an overview of stochastic calculus, with the Ito calculus chosen to show how to generalize ordinary calculus to the stochastic realm. The authors motivate the subject as one in which the functional form of stochastic processes was emphasized, with Ito attempting to find out just when local properties such as the drift and diffusion coefficients can characterize the stochastic process. The Ito formula is shown to be a generalization of the chain rule of ordinary calculus to the case where stochasticity is present. The authors are also careful to distinguish between "random" differential equations and "stochastic" differential equations. The former can be solved by integrating over differentiable sample paths, but in the latter one has to face the nondifferentiability of the sample paths, and hence solutions are more difficult to obtain. The authors give many examples of SDEs that can be solved explicitly, and prove existence and uniqueness theorems for strong solutions of the SDEs. And since ordinary differential equations are usually tackled by Taylor series expansions, it is perhaps not surprising that this technique would be generalized to SDEs, which the authors do in detail in this part. They also outline the differences between the Ito and Stratonovich interpretations of stochastic integrals and SDEs.

Part 3 is definitely of great interest to those who must develop mathematical models using SDEs. The authors carefully outline the reasons where Ito versus the Stratonovich formulations are used, this being largely dependent on the degree of autocorrelation in the processes at hand. The Stratonovich SDE is recommended for cases when the white noise is used as an idealization of a (smooth) real noise process. The authors also show how to approximate Markov chain problems with diffusion processes, which are the solutions of Ito SDEs. Several very interesting examples are given of the applications of stochastic differential equations; the particular ones of direct interest to me were the ones on population dynamics, protein kinetics, and genetics; option pricing, and blood clotting dynamics/cellular energetics.

After a review of discrete time approzimations in ordinary deterministic differential equations, in part 4 the authors show to solve SDEs using this approximation. The familiar Euler approximation is considered, with a simple example having an explicit solution compared with its Euler approximate solution. They also show how to use simulations when an explicit solution is lacking. The importance notions of strong and weak convergence ofthe approximate solutions are discussed in detail. Strong convergence is basically a convergence in norm (absolute value), while weak convergence is taken with respect to a collection of test functions. Both of these types of convergence reduce to the ordinary deterministic sense of convergence when the random elements are removed.

The discussion of convergence in part 4 leads to a very extensive discussion of strongly convergent approximations in part 5, and weakly convergent approximations in part 6. Stochastic Taylor expansions done with respect to the strong convergence criterion are discussed, beginning with the Euler approximation. More complicated strongly convergent stochastic approximation schemes are also considered, such as the Milstein scheme, which reduces to the Euler scheme when the diffusion coefficients only depend on time. The strong Taylor schemes of all orders are treated in detail. Since Taylor approximations make evaluations of the derivatives necessary, which is computational intensive, the authors discuss strong approximation schemes that do not require this, much like the Runge-Kutta methods in the deterministic case , but the authors are careful to point out that the Runge-Kutta analogy is problematic in the stochastic case. Several ofthese "derivative-free" schemes are considered by the authors. The authors also consider implicit strong approximation schemes for stiff SDEs, wherein numerical instabilities are problematic. Interesting applications are given for strong approximations for SDEs, such as the Duffing-Van der Pol oscillator, which is very important system in engineering mechanics and phyics, and has been subjected to an incredible amount of research.

More detailed consideration of weak Taylor approximations is given in part 6. The Euler scheme is examined first in the weak approximation, with the higher-order schemes following. Since weak convergence is more stringent than strong convergence, it should come as no surprise that fewer terms are required to obtain convergence, as compared with strong convergence at the same order. This intuition is indeed verified in the discussion, and the authors treat both explicit and implicit weak approximations, along with extrapolation and predictor-corrector methods. And most importantly, the authors give an introduction to the Girsanov methods for variance reduction of weak approximations to Ito diffusions, along with other techniques for doing the same. Those readers involved in constructive quantum field theory will value the treatment on using weak approximations to calculate functional integrals. The approximation of Lyapunov exponents for stochastic dynamical systems is also treated, along with the approximation of invariant measures. ... Read more

Product Description
This book gives a self-contained introduction to the subject of asymptotic approximation for multivariate integrals for both mathematicians and applied scientists. A collection of results of the Laplace methods is given. Such methods are useful for example in reliability, statistics, theoretical physics and information theory. An important special case is the approximation of multidimensional normal integrals. Here the relation between the differential geometry of the boundary of the integration domain and the asymptotic probability content is derived. One of the most important applications of these methods is in structural reliability. Engineers working in this field will find here a complete outline of asymptotic approximation methods for failure probability integrals. ... Read more

Product Description
This book is a survey of asymptotic methods set in the current applied research context of wave propagation. It stresses rigorous analysis in addition to formal manipulations. Asymptotic expansions developed in the text are justified rigorously, and students are shown how to obtain solid error estimates for asymptotic formulae. The book relates examples and exercises to subjects of current research interest, such as the problem of locating the zeros of Taylor polynomials of entire nonvanishing functions and the problem of counting integer lattice points in subsets of the plane with various geometrical properties of the boundary. The book is intended for a beginning graduate course on asymptotic analysis in applied mathematics and is aimed at students of pure and applied mathematics as well as science and engineering. The basic prerequisite is a background in differential equations, linear algebra, advanced calculus, and complex variables at the level of introductory undergraduate courses on these subjects. ... Read more

Product Description
Special functions arise in many problems of pure and applied mathematics, statistics, physics, and engineering. This book provides an up-to-date overview of methods for computing special functions and discusses when to use them in standard parameter domains, as well as in large and complex domains.

The first part of the book covers convergent and divergent series, Chebyshev expansions, numerical quadrature, and recurrence relations. Its focus is on the computation of special functions. Pseudoalgorithms are given to help students write their own algorithms. In addition to these basic tools, the authors discuss methods for computing zeros of special functions, uniform asymptotic expansions, Padé approximations, and sequence transformations. The book also provides specific algorithms for computing several special functions (Airy functions and parabolic cylinder functions, among others).

Audience: This book is intended for researchers in applied mathematics, scientific computing, physics, engineering, statistics, and other scientific disciplines in which special functions are used as computational tools. Some chapters can be used in general numerical analysis courses.

Contents: List of Algorithms; Preface; Chapter 1: Introduction; Part I: Basic Methods. Chapter 2: Convergent and Divergent Series; Chapter 3: Chebyshev Expansions; Chapter 4: Linear Recurrence Relations and Associated Continued Fractions; Chapter 5: Quadrature Methods; Part II: Further Tools and Methods. Chapter 6: Numerical Aspects of Continued Fractions; Chapter 7: Computation of the Zeros of Special Functions; Chapter 8: Uniform Asymptotic Expansions; Chapter 9: Other Methods; Part III: Related Topics and Examples. Chapter 10: Inversion of Cumulative Distribution Functions; Chapter 11: Further Examples; Part IV: Software. Chapter 12: Associated Algorithms; Bibliography; Index. ... Read more

Product Description

The book provides an easily accessible computationally oriented introduction into the numerical solution of stochastic differential equations using computer experiments. It develops in the reader an ability to apply numerical methods solving stochastic differential equations in their own fields. Furthermore, it creates an intuitive understanding of the necessary theoretical background from stochastic and numeric analysis.  A downloadable softward containing programs for over 100 problems is provided at each of the following homepages:

http://www.math.uni-frankfurt.de/~numerik/kloeden/
http://www.business.uts.edu.au/finance/staff/eckhard.html
http.//www.math.siu.edu/schurz/SOFTWARE/

to enable the reader to develop an intuitive understanding of the issues involved. Applications include stochastic dynamical systems, filtering, parametric estimation and finance modeling.

The book is intended for readers without specialist stochastic background who want to apply such numerical methods to stochastic differential equations that arise in their own filed.

Product Description
Structural stability is of fundamental importance inmaterials science. Up-to-date information on the theoreticalaspects of phase stability of materials is contained in thisvolume. Most of the first-principles calculations are basedon the local-density approximation (LDA). In contrast, thisvolume contains very recent results of "going beyond LDA",such as the density gradient expansion and the quantumMonte-Carlomethod.Following the recently introduced theoretical methods forthe calculation of interatomic potentials, forces acting onatoms and total energies such as the Car-Parrinello, theeffective-medium and the bond-ordermethod, attempts havebeen made to develop even more sophisticated methodssuch asthe order-N method in electronic-structure calculations.Thepresent status of these methods and their application toreal systems are described.In addition, in order to study the phase stability atfinitetemperatures, the microscopic calculations have to becombined with statistical treatment of the systems todescribe, e.g. order-disordertransitions on the Si(001)surface or alloy phase diagrams. This book contains examplesfor this type of calculations. ... Read more

Product Description
If you want top grades and thorough understanding of partial differential equations, this powerful study tool is the best tutor you can have! It takes you step-by-step through the subject and gives you 290 accompanying related problems with fully worked solutions. You also get plenty of practice problems to do on your own, working at your own speed. (Answers at the back show you how you're doing.) Famous for their clarity, wealth of illustrations and examples, and lack of dreary minutiae, SchaumOs Outlines have sold more than 30 million copies worldwideNand this guide will show you why! ... Read more

Customer Reviews (14)

new book

I am satisfy with the new book which I received from my collegue who is going to San Jose for training last month.

Good book, easy to understand
This series is very famous , and i like the methods and way they interprete.

Partial Differential Equations review
I have found it helpful and inciteful with the more difficult differential equations that can be attempted.

Good if you've forgotten
This book contains mostly routine exercises of the subject. If you want to dig a bit further and sharpen your skills: try Krasnov's A Book of Problems in Ordinary Differential Equations. Language: English
ISBN: 5030009493

Not up to par with other Schaum's outlines on mathematics
This is one of the more poorly written Schaum's outlines I have encountered. The theory is very murky and the author gives no clear direction as to where he is going with this material and what it all means. PDE is a hard enough subject without working a bunch of meaningless problems that leave you wondering what it is you are supposed to have learned. Instead, I suggest you read "Introduction to Partial Differential Equations with Applications" by Zachmanoglou and Thoe in order to understand the mathematical underpinnings of PDE. Then read "Partial Differential Equations for Scientists and Engineers" to get a thorough feel for how PDE is used to solve real-world problems. Both books usually sell used for under $10 each, making them cost-effective alternatives to this Schaum's outline. ... Read more