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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.
Industry Reviews
| Preface | p. VII |
| Notation | p. XVII |
| Stochastic Processes and Stochastic Differential Equations | p. 1 |
| Elements of probability and random variables | p. 1 |
| Mean, variance, and moments | p. 2 |
| Change of measure and Radon-Nikodym derivative | p. 4 |
| Random number generation | p. 5 |
| The Monte Carlo method | p. 5 |
| Variance reduction techniques | p. 8 |
| Preferential sampling | p. 9 |
| Control variables | p. 12 |
| Antithetic sampling | p. 13 |
| Generalities of stochastic processes | p. 14 |
| Filtrations | p. 14 |
| Simple and quadratic variation of a process | p. 15 |
| Moments, covariance, and increments of stochastic processes | p. 16 |
| Conditional expectation | p. 16 |
| Martingales | p. 18 |
| Brownian motion | p. 18 |
| Brownian motion as the limit of a random walk | p. 20 |
| Brownian motion as L[superscript 2 0, T] expansion | p. 22 |
| Brownian motion paths are nowhere differentiable | p. 24 |
| Geometric Brownian motion | p. 24 |
| Brownian bridge | p. 27 |
| Stochastic integrals and stochastic differential equations | p. 29 |
| Properties of the stochastic integral and Ito processes | p. 32 |
| Diffusion processes | p. 33 |
| Ergodicity | p. 35 |
| Markovianity | p. 36 |
| Quadratic variation | p. 37 |
| Infinitesimal generator of a diffusion process | p. 37 |
| How to obtain a martingale from a diffusion process | p. 37 |
| Ito formula | p. 38 |
| Orders of differentials in the Ito formula | p. 38 |
| Linear stochastic differential equations | p. 39 |
| Derivation of the SDE for the geometric Brownian motion | p. 39 |
| The Lamperti transform | p. 40 |
| Girsanov's theorem and likelihood ratio for diffusion processes | p. 41 |
| Some parametric families of stochastic processes | p. 43 |
| Ornstein-Uhlenbeck or Vasicek process | p. 43 |
| The Black-Scholes-Merton or geometric Brownian motion model | p. 46 |
| The Cox-Ingersoll-Ross model | p. 47 |
| The CKLS family of models | p. 49 |
| The modified CIR and hyperbolic processes | p. 49 |
| The hyperbolic processes | p. 50 |
| The nonlinear mean reversion Ait-Sahalia model | p. 50 |
| Double-well potential | p. 51 |
| The Jacobi diffusion process | p. 51 |
| Ahn and Gao model or inverse of Feller's square root model | p. 52 |
| Radial Ornstein-Uhlenbeck process | p. 52 |
| Pearson diffusions | p. 52 |
| Another classification of linear stochastic systems | p. 54 |
| One epidemic model | p. 56 |
| The stochastic cusp catastrophe model | p. 57 |
| Exponential families of diffusions | p. 58 |
| Generalized inverse gaussian diffusions | p. 59 |
| Numerical Methods for SDE | p. 61 |
| Euler approximation | p. 62 |
| A note on code vectorization | p. 63 |
| Milstein scheme | p. 65 |
| Relationship between Milstein and Euler schemes | p. 66 |
| Transform of the geometric Brownian motion | p. 68 |
| Transform of the Cox-Ingersoll-Ross process | p. 68 |
| Implementation of Euler and Milstein schemes: the sde.sim function | p. 69 |
| Example of use | p. 70 |
| The constant elasticity of variance process and strange paths | p. 72 |
| Predictor-corrector method | p. 72 |
| Strong convergence for Euler and Milstein schemes | p. 74 |
| KPS method of strong order [gamma] = 1.5 | p. 77 |
| Second Milstein scheme | p. 81 |
| Drawing from the transition density | p. 82 |
| The Ornstein-Uhlenbeck or Vasicek process | p. 83 |
| The Black and Scholes process | p. 83 |
| The CIR process | p. 83 |
| Drawing from one model of the previous classes | p. 84 |
| Local linearization method | p. 85 |
| The Ozaki method | p. 85 |
| The Shoji-Ozaki method | p. 87 |
| Exact sampling | p. 91 |
| Simulation of diffusion bridges | p. 98 |
| The algorithm | p. 99 |
| Numerical considerations about the Euler scheme | p. 101 |
| Variance reduction techniques | p. 102 |
| Control variables | p. 103 |
| Summary of the function sde.sim | p. 105 |
| Tips and tricks on simulation | p. 106 |
| Parametric Estimation | p. 109 |
| Exact likelihood inference | p. 112 |
| The Ornstein-Uhlenbeck or Vasicek model | p. 113 |
| The Black and Scholes or geometric Brownian motion model | p. 117 |
| The Cox-Ingersoll-Ross model | p. 119 |
| Pseudo-likelihood methods | p. 122 |
| Euler method | p. 122 |
| Elerian method | p. 125 |
| Local linearization methods | p. 127 |
| Comparison of pseudo-likelihoods | p. 128 |
| Approximated likelihood methods | p. 131 |
| Kessler method | p. 131 |
| Simulated likelihood method | p. 134 |
| Hermite polynomials expansion of the likelihood | p. 138 |
| Bayesian estimation | p. 155 |
| Estimating functions | p. 157 |
| Simple estimating functions | p. 157 |
| Algorithm 1 for simple estimating functions | p. 164 |
| Algorithm 2 for simple estimating functions | p. 167 |
| Martingale estimating functions | p. 172 |
| Polynomial martingale estimating functions | p. 173 |
| Estimating functions based on eigenfunctions | p. 178 |
| Estimating functions based on transform functions | p. 179 |
| Discretization of continuous-time estimators | p. 179 |
| Generalized method of moments | p. 182 |
| The GMM algorithm | p. 184 |
| GMM, stochastic differential equations, and Euler method | p. 185 |
| What about multidimensional diffusion processes? | p. 190 |
| Miscellaneous Topics | p. 191 |
| Model identification via Akaike's information criterion | p. 191 |
| Nonparametric estimation | p. 197 |
| Stationary density estimation | p. 198 |
| Local-time and stationary density estimators | p. 201 |
| Estimation of diffusion and drift coefficients | p. 202 |
| Change-point estimation | p. 208 |
| Estimation of the change point with unknown drift | p. 212 |
| A famous example | p. 215 |
| A brief excursus into R | p. 217 |
| Typing into the R console | p. 217 |
| Assignments | p. 218 |
| R vectors and linear algebra | p. 220 |
| Subsetting | p. 221 |
| Different types of objects | p. 222 |
| Expressions and functions | p. 225 |
| Loops and vectorization | p. 227 |
| Environments | p. 228 |
| Time series objects | p. 229 |
| R Scripts | p. 231 |
| Miscellanea | p. 232 |
| The sde Package | p. 233 |
| BM | p. 234 |
| cpoint | p. 235 |
| DBridge | p. 236 |
| dcElerian | p. 237 |
| dcEuler | p. 238 |
| dcKessler | p. 238 |
| dcOzaki | p. 239 |
| dcShoji | p. 240 |
| dcSim | p. 241 |
| DWJ | p. 243 |
| EULERloglik | p. 243 |
| gmm | p. 245 |
| HPloglik | p. 247 |
| ksmooth | p. 248 |
| linear.mart.ef | p. 250 |
| rcBS | p. 251 |
| rcCIR | p. 252 |
| rcOU | p. 253 |
| rsCIR | p. 254 |
| rsOU | p. 255 |
| sde.sim | p. 256 |
| sdeAIC | p. 259 |
| SIMloglik | p. 261 |
| simple.ef | p. 262 |
| simple.ef2 | p. 264 |
| References | p. 267 |
| Index | p. 279 |
| Table of Contents provided by Ingram. All Rights Reserved. |
ISBN: 9780387758381
ISBN-10: 0387758380
Series: Springer Series in Statistics
Published: 5th May 2008
Format: Hardcover
Language: English
Number of Pages: 304
Audience: General Adult
Publisher: Springer Nature B.V.
Country of Publication: GB
Dimensions (cm): 23.8 x 16.2 x 1.9
Weight (kg): 0.56
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- Non-FictionMathematicsApplied MathematicsStochastics
- Non-FictionMathematicsCalculus & Mathematical AnalysisDifferential Calculus & Equations
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