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Simulation and Inference for Stochastic Differential Equations : With R Examples - Stefano M. Iacus

Simulation and Inference for Stochastic Differential Equations

With R Examples


Published: 5th May 2008
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This book is unique because of its focus on the practical implementation of the simulation and estimation methods presented. The book will be useful to practitioners and students with only a minimal mathematical background because of the many R programs, and to more mathematically-educated practitioners.

Many of the methods presented in the book have 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 into 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 publications. The third one focuses 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 point estimation. 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. A documentation page is available at the end of the book for each R function presented in the book.

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 stochastic 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 stochastic 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 stochastic 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)

Prefacep. VII
Notationp. XVII
Stochastic Processes and Stochastic Differential Equationsp. 1
Elements of probability and random variablesp. 1
Mean, variance, and momentsp. 2
Change of measure and Radon-Nikodym derivativep. 4
Random number generationp. 5
The Monte Carlo methodp. 5
Variance reduction techniquesp. 8
Preferential samplingp. 9
Control variablesp. 12
Antithetic samplingp. 13
Generalities of stochastic processesp. 14
Filtrationsp. 14
Simple and quadratic variation of a processp. 15
Moments, covariance, and increments of stochastic processesp. 16
Conditional expectationp. 16
Martingalesp. 18
Brownian motionp. 18
Brownian motion as the limit of a random walkp. 20
Brownian motion as L[superscript 2 0, T] expansionp. 22
Brownian motion paths are nowhere differentiablep. 24
Geometric Brownian motionp. 24
Brownian bridgep. 27
Stochastic integrals and stochastic differential equationsp. 29
Properties of the stochastic integral and Ito processesp. 32
Diffusion processesp. 33
Ergodicityp. 35
Markovianityp. 36
Quadratic variationp. 37
Infinitesimal generator of a diffusion processp. 37
How to obtain a martingale from a diffusion processp. 37
Ito formulap. 38
Orders of differentials in the Ito formulap. 38
Linear stochastic differential equationsp. 39
Derivation of the SDE for the geometric Brownian motionp. 39
The Lamperti transformp. 40
Girsanov's theorem and likelihood ratio for diffusion processesp. 41
Some parametric families of stochastic processesp. 43
Ornstein-Uhlenbeck or Vasicek processp. 43
The Black-Scholes-Merton or geometric Brownian motion modelp. 46
The Cox-Ingersoll-Ross modelp. 47
The CKLS family of modelsp. 49
The modified CIR and hyperbolic processesp. 49
The hyperbolic processesp. 50
The nonlinear mean reversion Ait-Sahalia modelp. 50
Double-well potentialp. 51
The Jacobi diffusion processp. 51
Ahn and Gao model or inverse of Feller's square root modelp. 52
Radial Ornstein-Uhlenbeck processp. 52
Pearson diffusionsp. 52
Another classification of linear stochastic systemsp. 54
One epidemic modelp. 56
The stochastic cusp catastrophe modelp. 57
Exponential families of diffusionsp. 58
Generalized inverse gaussian diffusionsp. 59
Numerical Methods for SDEp. 61
Euler approximationp. 62
A note on code vectorizationp. 63
Milstein schemep. 65
Relationship between Milstein and Euler schemesp. 66
Transform of the geometric Brownian motionp. 68
Transform of the Cox-Ingersoll-Ross processp. 68
Implementation of Euler and Milstein schemes: the sde.sim functionp. 69
Example of usep. 70
The constant elasticity of variance process and strange pathsp. 72
Predictor-corrector methodp. 72
Strong convergence for Euler and Milstein schemesp. 74
KPS method of strong order [gamma] = 1.5p. 77
Second Milstein schemep. 81
Drawing from the transition densityp. 82
The Ornstein-Uhlenbeck or Vasicek processp. 83
The Black and Scholes processp. 83
The CIR processp. 83
Drawing from one model of the previous classesp. 84
Local linearization methodp. 85
The Ozaki methodp. 85
The Shoji-Ozaki methodp. 87
Exact samplingp. 91
Simulation of diffusion bridgesp. 98
The algorithmp. 99
Numerical considerations about the Euler schemep. 101
Variance reduction techniquesp. 102
Control variablesp. 103
Summary of the function sde.simp. 105
Tips and tricks on simulationp. 106
Parametric Estimationp. 109
Exact likelihood inferencep. 112
The Ornstein-Uhlenbeck or Vasicek modelp. 113
The Black and Scholes or geometric Brownian motion modelp. 117
The Cox-Ingersoll-Ross modelp. 119
Pseudo-likelihood methodsp. 122
Euler methodp. 122
Elerian methodp. 125
Local linearization methodsp. 127
Comparison of pseudo-likelihoodsp. 128
Approximated likelihood methodsp. 131
Kessler methodp. 131
Simulated likelihood methodp. 134
Hermite polynomials expansion of the likelihoodp. 138
Bayesian estimationp. 155
Estimating functionsp. 157
Simple estimating functionsp. 157
Algorithm 1 for simple estimating functionsp. 164
Algorithm 2 for simple estimating functionsp. 167
Martingale estimating functionsp. 172
Polynomial martingale estimating functionsp. 173
Estimating functions based on eigenfunctionsp. 178
Estimating functions based on transform functionsp. 179
Discretization of continuous-time estimatorsp. 179
Generalized method of momentsp. 182
The GMM algorithmp. 184
GMM, stochastic differential equations, and Euler methodp. 185
What about multidimensional diffusion processes?p. 190
Miscellaneous Topicsp. 191
Model identification via Akaike's information criterionp. 191
Nonparametric estimationp. 197
Stationary density estimationp. 198
Local-time and stationary density estimatorsp. 201
Estimation of diffusion and drift coefficientsp. 202
Change-point estimationp. 208
Estimation of the change point with unknown driftp. 212
A famous examplep. 215
A brief excursus into Rp. 217
Typing into the R consolep. 217
Assignmentsp. 218
R vectors and linear algebrap. 220
Subsettingp. 221
Different types of objectsp. 222
Expressions and functionsp. 225
Loops and vectorizationp. 227
Environmentsp. 228
Time series objectsp. 229
R Scriptsp. 231
Miscellaneap. 232
The sde Packagep. 233
BMp. 234
cpointp. 235
DBridgep. 236
dcElerianp. 237
dcEulerp. 238
dcKesslerp. 238
dcOzakip. 239
dcShojip. 240
dcSimp. 241
DWJp. 243
EULERloglikp. 243
gmmp. 245
HPloglikp. 247
ksmoothp. 248
linear.mart.efp. 250
rcBSp. 251
rcCIRp. 252
rcOUp. 253
rsCIRp. 254
rsOUp. 255
sde.simp. 256
sdeAICp. 259
SIMloglikp. 261
simple.efp. 262
simple.ef2p. 264
Referencesp. 267
Indexp. 279
Table of Contents provided by Ingram. All Rights Reserved.

ISBN: 9780387758381
ISBN-10: 0387758380
Series: Springer Series in Statistics
Audience: Tertiary; University or College
Format: Hardcover
Language: English
Number Of Pages: 285
Published: 5th May 2008
Publisher: Springer-Verlag New York Inc.
Country of Publication: US
Dimensions (cm): 23.5 x 15.5  x 1.91
Weight (kg): 0.56