Simulation and Inference for Stochastic Differential Equations: With R Examples / Edition 1

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Overview

This book is very different from any other publication in the field and it is unique because of its focus on the practical implementation of the simulation and estimation methods presented. The book should be useful to practitioners and students with minimal mathematical background, but because of the many R programs, probably also to many mathematically well educated practitioners. Many of the methods presented in the book have, so far, not been used much in practice because the lack of an implementation in a unified framework. This book fills the gap. With the R code included in this book, a lot of useful methods become easy to use for practitioners and students. An R package called 'sde' provides functions with easy interfaces ready to be used on empirical data from real life applications. Although it contains a wide range of results, the book has an introductory character and necessarily does not cover the whole spectrum of simulation and inference for general stochastic differential equations.

The book is organized in four chapters. The first one introduces the subject and presents several classes of processes used in many fields of mathematics, computational biology, finance and the social sciences. The second chapter is devoted to simulation schemes and covers new methods not available in other milestones publication known so far. The third one is focused on parametric estimation techniques. In particular, it includes exact likelihood inference, approximated and pseudo-likelihood methods, estimating functions, generalized method of moments and other techniques. The last chapter contains miscellaneous topics like nonparametric estimation, model identification and change pointestimation. The reader who is not an expert in the R-language will find a concise introduction to this environment focused on the subject of the book which should allow for instant use of the proposed material. A documentation page is available at the end of the book for each R function presented in the book.

About the Author:
Stefano M. Iacus is associate professor of Probability and Mathematical Statistics at the University of Milan, Department of Economics, Business and Statistics

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Editorial Reviews

From the Publisher
From the reviews:

"It is a pleasure to strongly recommend the text to the intended audience.The writing style is effective, with a relatively gentle but accurate mathematicalcoverage and a wealth of R code in the sde package." (Thomas L. Burr, Technometrics, V51, N3)

"The book focuses on simulation techniques and parameter estimation for SDEs. With the examples is included a detailed program code in R.It is written in a way so that it is suitable for (1) the beginner who meets shastic differential equations (SDEs) for the first time and needs to do simulation or estimation and (2) the advanced reader who wants to know about new directions on numerics or inference and already knows the standard theory.… There is also an interesting small chapter on miscellaneous topics which contains the Akaike information criterion, non-parametric estimation and change-point estimation. Essentially all examples are complemented by program codes in R. The last chapter focuses on aspects of the language that are used throughout the book. Generally the codes are, without much effort, translatable into other languages." (Roger Pettersson, American Mathematical Society 2009, MR2410254 (Review) 60H10 (62F10 65C30))

"This book succeeds at giving an overview of a complicated topic through a mix of simplified theory and examples, while pointing the reader in the right direction for more information.… This would be a good introductory or reference text for a graduate level course, where the instructor’s knowledge extends substantially beyond the book.… data examples are abundant and give the book the feeling of being practical while showcasing when methods succeed and fail." (Dave Cambell, Biometrics, 65, 326-339, March 2009)

"Overall, this is a good book that fills in several gaps. In addition to collecting and summarizing an enormous quantity of theory, it introduces some novel techniques for inference. Statisticians and mathemeticians who work with time series should find a place on their shelves for this book." (Journal of Statistical Software - Book Reviews)

"Diffusion processes, described by shastic differential equations, are extensively applied in many areas of scientific research. There are many books of the subject with emphasis on either theory of applications. However, there is not much literature available on practical implementation of these models. Therefore, this book is welcome and helps fill a gap. … the thorough coverage of univariate models provided by the book is also useful. These models are building blocks for larger models, and it is good to have a handy reference to their properties, such as parameter restrictions and stationary distributions." (Arto Luoma, (International Statistical Review, 2009, 77, 1)

"In summary, this book is indeed quite unique: it gives a concise methodological survey with strong focus on applications and provides many ready-to-use recipes. The theory is always illustrated with detailed examples incorporating various parametric diffusion models. This text is a recommended acquisition for practitioners both in the industry and in applied disciplines of academia." (Marco Frei, ETH Zurich, JASA March2010, v105(489)

"To summarize, this book fills several gaps in the literature, summarizing the theory of sto- chastic processes and introducing some new estimation techniques. The main strength of the book is the breadth of its scope. It covers the basic theory of the shastic processes, appli- cations, an implementation in concrete com- puter codes. An empirical economist would find Chapter 3 most important, while for a theorist it will be useful to concentrate on Chapter 1." (Suren Basov, La Trobe University, Economic Records, v86(272), March 2010)

“…This book is indeed quite unique; it gives a concise methodological survey with strong focus on applications and provides many ready-to- use recipes. The theory is always illustrated with detailed examples incorporating various parametric diffusion models. This text is a recommended acquisition for practitioners both in the industry and in applied disciplines of academia.” (Journal of the American Statistical Association, Vol. 105, No. 489)

“This book focuses on simulation techniques and parameter estimation for SDEs. It gives an overview of these topics through a mix of simplified theory and examples. The book is written in a way to be suitable for the beginner and the advanced reader who want to know about new directions in numerics or inference.” (Rainer Schlittgen, Zentralblatt MATH, Vol. 1210, 2011)

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Product Details

  • ISBN-13: 9780387758381
  • Publisher: Springer New York
  • Publication date: 5/30/2008
  • Series: Springer Series in Statistics
  • Edition description: 2008
  • Edition number: 1
  • Pages: 286
  • Product dimensions: 6.30 (w) x 9.30 (h) x 0.80 (d)

Table of Contents


Preface     VII
Notation     XVII
Stochastic Processes and Stochastic Differential Equations     1
Elements of probability and random variables     1
Mean, variance, and moments     2
Change of measure and Radon-Nikodym derivative     4
Random number generation     5
The Monte Carlo method     5
Variance reduction techniques     8
Preferential sampling     9
Control variables     12
Antithetic sampling     13
Generalities of stochastic processes     14
Filtrations     14
Simple and quadratic variation of a process     15
Moments, covariance, and increments of stochastic processes     16
Conditional expectation     16
Martingales     18
Brownian motion     18
Brownian motion as the limit of a random walk     20
Brownian motion as L[superscript 2 0, T] expansion     22
Brownian motion paths are nowhere differentiable     24
Geometric Brownian motion     24
Brownian bridge     27
Stochastic integrals and stochastic differential equations     29
Properties of the stochastic integral and Ito processes     32
Diffusion processes     33
Ergodicity     35
Markovianity     36
Quadratic variation     37
Infinitesimal generator of a diffusion process     37
How to obtain a martingale from a diffusion process     37
Ito formula     38
Orders of differentials in the Ito formula     38
Linear stochastic differential equations     39
Derivation of the SDE for the geometric Brownian motion     39
The Lamperti transform     40
Girsanov's theorem and likelihood ratio for diffusion processes     41
Some parametric families of stochastic processes     43
Ornstein-Uhlenbeck or Vasicek process     43
The Black-Scholes-Merton or geometric Brownian motion model     46
The Cox-Ingersoll-Ross model     47
The CKLS family of models     49
The modified CIR and hyperbolic processes     49
The hyperbolic processes     50
The nonlinear mean reversion Ait-Sahalia model     50
Double-well potential     51
The Jacobi diffusion process     51
Ahn and Gao model or inverse of Feller's square root model     52
Radial Ornstein-Uhlenbeck process      52
Pearson diffusions     52
Another classification of linear stochastic systems     54
One epidemic model     56
The stochastic cusp catastrophe model     57
Exponential families of diffusions     58
Generalized inverse gaussian diffusions     59
Numerical Methods for SDE     61
Euler approximation     62
A note on code vectorization     63
Milstein scheme     65
Relationship between Milstein and Euler schemes     66
Transform of the geometric Brownian motion     68
Transform of the Cox-Ingersoll-Ross process     68
Implementation of Euler and Milstein schemes: the sde.sim function     69
Example of use     70
The constant elasticity of variance process and strange paths     72
Predictor-corrector method     72
Strong convergence for Euler and Milstein schemes     74
KPS method of strong order [gamma] = 1.5     77
Second Milstein scheme     81
Drawing from the transition density     82
The Ornstein-Uhlenbeck or Vasicek process     83
The Black and Scholes process     83
The CIR process     83
Drawing from one model of the previous classes     84
Local linearization method     85
The Ozaki method     85
The Shoji-Ozaki method     87
Exact sampling     91
Simulation of diffusion bridges     98
The algorithm     99
Numerical considerations about the Euler scheme     101
Variance reduction techniques     102
Control variables     103
Summary of the function sde.sim     105
Tips and tricks on simulation     106
Parametric Estimation     109
Exact likelihood inference     112
The Ornstein-Uhlenbeck or Vasicek model     113
The Black and Scholes or geometric Brownian motion model     117
The Cox-Ingersoll-Ross model     119
Pseudo-likelihood methods     122
Euler method     122
Elerian method     125
Local linearization methods     127
Comparison of pseudo-likelihoods     128
Approximated likelihood methods     131
Kessler method     131
Simulated likelihood method     134
Hermite polynomials expansion of the likelihood     138
Bayesian estimation     155
Estimating functions      157
Simple estimating functions     157
Algorithm 1 for simple estimating functions     164
Algorithm 2 for simple estimating functions     167
Martingale estimating functions     172
Polynomial martingale estimating functions     173
Estimating functions based on eigenfunctions     178
Estimating functions based on transform functions     179
Discretization of continuous-time estimators     179
Generalized method of moments     182
The GMM algorithm     184
GMM, stochastic differential equations, and Euler method     185
What about multidimensional diffusion processes?     190
Miscellaneous Topics     191
Model identification via Akaike's information criterion     191
Nonparametric estimation     197
Stationary density estimation     198
Local-time and stationary density estimators     201
Estimation of diffusion and drift coefficients     202
Change-point estimation     208
Estimation of the change point with unknown drift     212
A famous example     215
A brief excursus into R     217
Typing into the R console     217
Assignments      218
R vectors and linear algebra     220
Subsetting     221
Different types of objects     222
Expressions and functions     225
Loops and vectorization     227
Environments     228
Time series objects     229
R Scripts     231
Miscellanea     232
The sde Package     233
BM     234
cpoint     235
DBridge     236
dcElerian     237
dcEuler     238
dcKessler     238
dcOzaki     239
dcShoji     240
dcSim     241
DWJ     243
EULERloglik     243
gmm     245
HPloglik     247
ksmooth     248
linear.mart.ef     250
rcBS     251
rcCIR     252
rcOU     253
rsCIR     254
rsOU     255
sde.sim     256
sdeAIC     259
SIMloglik     261
simple.ef     262
simple.ef2     264
References     267
Index      279
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