【经典】Numerical Solution of SDEs, 随机微分方程数值解 [推广有奖]

【推荐】目前为止最全面、最权威的随机微分方程数值解法的著作。搞金融、随机系统、生物数学的人都不该错过。
【理由】本论坛居然没有。。。
【书名】Numerical Solution of Stochastic Differential Equations, 随机微分方程数值解
【作者】Peter E. Kloeden (Author), Eckhard Platen (Author)
【格式】djvu，清晰
【简介】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.
【书评】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.
Aspreparation for the study of SDEs, the authors detail some preliminarybackground on probability, statistics, and stochastic processes in Part1 of the book. Particularly well-written is the discussion on randomnumber generators and efficient methods for generating random numbers,such as the Box-Muller and Polar Marsaglia methods. Both discrete andcontinuous Markov processes are discussed, and the authors review theconnection between Weiner processes (Brownian motion for the physicistreader) and white noise. The measure-theory foundations of the subjectare outlined briefly for the interested reader.
Part 2 beginsnaturally with an overview of stochastic calculus, with the Itocalculus chosen to show how to generalize ordinary calculus to thestochastic realm. The authors motivate the subject as one in which thefunctional form of stochastic processes was emphasized, with Itoattempting to find out just when local properties such as the drift anddiffusion coefficients can characterize the stochastic process. The Itoformula is shown to be a generalization of the chain rule of ordinarycalculus to the case where stochasticity is present. The authors arealso careful to distinguish between "random" differential equations and"stochastic" differential equations. The former can be solved byintegrating over differentiable sample paths, but in the latter one hasto face the nondifferentiability of the sample paths, and hencesolutions are more difficult to obtain. The authors give many examplesof SDEs that can be solved explicitly, and prove existence anduniqueness theorems for strong solutions of the SDEs. And sinceordinary differential equations are usually tackled by Taylor seriesexpansions, it is perhaps not surprising that this technique would begeneralized to SDEs, which the authors do in detail in this part. Theyalso outline the differences between the Ito and Stratonovichinterpretations of stochastic integrals and SDEs.
Part 3 isdefinitely of great interest to those who must develop mathematicalmodels using SDEs. The authors carefully outline the reasons where Itoversus the Stratonovich formulations are used, this being largelydependent on the degree of autocorrelation in the processes at hand.The Stratonovich SDE is recommended for cases when the white noise isused as an idealization of a (smooth) real noise process. The authorsalso show how to approximate Markov chain problems with diffusionprocesses, which are the solutions of Ito SDEs. Several veryinteresting examples are given of the applications of stochasticdifferential equations; the particular ones of direct interest to mewere the ones on population dynamics, protein kinetics, and genetics;option pricing, and blood clotting dynamics/cellular energetics.
Aftera review of discrete time approzimations in ordinary deterministicdifferential equations, in part 4 the authors show to solve SDEs usingthis approximation. The familiar Euler approximation is considered,with a simple example having an explicit solution compared with itsEuler approximate solution. They also show how to use simulations whenan explicit solution is lacking. The importance notions of strong andweak convergence of the 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 testfunctions. Both of these types of convergence reduce to the ordinarydeterministic sense of convergence when the random elements areremoved.
The discussion of convergence in part 4 leads to avery extensive discussion of strongly convergent approximations in part5, and weakly convergent approximations in part 6. Stochastic Taylorexpansions done with respect to the strong convergence criterion arediscussed, beginning with the Euler approximation. More complicatedstrongly convergent stochastic approximation schemes are alsoconsidered, such as the Milstein scheme, which reduces to the Eulerscheme when the diffusion coefficients only depend on time. The strongTaylor schemes of all orders are treated in detail. Since Taylorapproximations make evaluations of the derivatives necessary, which iscomputational intensive, the authors discuss strong approximationschemes that do not require this, much like the Runge-Kutta methods inthe deterministic case , but the authors are careful to point out thatthe Runge-Kutta analogy is problematic in the stochastic case. Severalof these "derivative-free" schemes are considered by the authors. Theauthors also consider implicit strong approximation schemes for stiffSDEs, wherein numerical instabilities are problematic. Interestingapplications are given for strong approximations for SDEs, such as theDuffing-Van der Pol oscillator, which is very important system inengineering mechanics and phyics, and has been subjected to anincredible amount of research.
More detailed consideration ofweak Taylor approximations is given in part 6. The Euler scheme isexamined first in the weak approximation, with the higher-order schemesfollowing. Since weak convergence is more stringent than strongconvergence, it should come as no surprise that fewer terms arerequired to obtain convergence, as compared with strong convergence atthe 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 mostimportantly, the authors give an introduction to the Girsanov methodsfor variance reduction of weak approximations to Ito diffusions, alongwith other techniques for doing the same. Those readers involved inconstructive quantum field theory will value the treatment on usingweak approximations to calculate functional integrals. Theapproximation of Lyapunov exponents for stochastic dynamical systems isalso treated, along with the approximation of invariant measures.