Simulation and Inference for Stochastic Processes with Yuima: A Comprehensive R Framework for Sdes and Other Stochastic Processes

ISBN-13: 9783319555676 / Angielski / Miękka / 2018 / 268 str.

Stefano M. Iacus; Nakahiro Yoshida

Opis

The YUIMA package is the first comprehensive R framework based on S4 classes and methods which allows for the simulation of stochastic differential equations driven by Wiener process, Levy processes or fractional Brownian motion, as well as CARMA, COGARCH, and Point processes. The package performs various central statistical analyses such as quasi maximum likelihood estimation, adaptive Bayes estimation, structural change point analysis, hypotheses testing, asynchronous covariance estimation, lead-lag estimation, LASSO model selection, and so on. YUIMA also supports stochastic numerical analysis by fast computation of the expected value of functionals of stochastic processes through automatic asymptotic expansion by means of the Malliavin calculus. All models can be multidimensional, multiparametric or non parametric.The book explains briefly the underlying theory for simulation and inference of several classes of stochastic processes and then presents both simulation experiments and applications to real data. Although these processes have been originally proposed in physics and more recently in finance, they are becoming popular also in biology due to the fact the time course experimental data are now available. The YUIMA package, available on CRAN, can be freely downloaded and this companion book will make the user able to start his or her analysis from the first page.

Recenzje

"The book is written in an accessible style ... . a wide audience can benefit for a through overview of the topic and the R package." (Irina Ioana Mohorianu, zbMATH 1458.60004, 2021)

Spis treści

Introduction

1.1 Overview of the project

1.2 Who should read this book?

1.3 Structure of the book

1.4 How to get the R code for this book

1.5 Main contribution to the Yuima package

1.6 Further developments of Yuima Package

1.7 Things to know about R

1.7.1 How to get R

1.7.2 R and S4 objects

1.8 The yuima package

1.8.1 How to obtain the package

1.8.2 The main object and classes

1.8.3 The yuima.model class

1.9 On model specification

1.9.1 Basic model specification

1.9.2 User-specified state and time variables

1.9.3 Specification of parametric models

1.10 Basic facts on simulation

1.10.1 Customization of simulation arguments

1.10.2 Simulation of models with user-specified notation

1.10.3 Simulation of parametric models

1.11 Sampling and simulate

1.11.1 Sampling and subsampling

1.12 How to make data available into a yuima object

1.12.1 Getting data from data providers

1.13 How to extract data from a yuima object

1.14 Time series classes, time data and time stamps

1.14.1 Review of some time series objects in R

1.14.2 How to handle real time stamps

1.14.3 Dates manipulation

1.14.4 Using dates to index time series

1.14.5 Joining two or more time series

1.14.6 Subsetting a time series

1.15 Miscellanea

1.15.1 From Yuima to LATEX

1.15.2 The Yuima GUI

 

Part II Models and Inference

2 Diffusion processes

2.1 One dimensional model specification

2.1.1 Ornstein-Uhlenbeck (OU)

2.1.2 Geometric Brownian motion (gBm)

2.1.3 Vasicek model (VAS)

2.1.4 Constant elasticity of variance (CEV)

2.1.5 Cox-Ingersoll-Ross process (CIR)

2.1.6 Chan-Karolyi-Longstaff-Sanders process (CKLS)

2.1.7 Hyperbolic diffusion processes

2.2 More about simulation

2.3 Space-discretized Euler-Maruyama simulation scheme

2.4 Multidimensional processes

2.4.1 The Heston model

2.5 Parametric inference

2.5.1 Quasi maximum likelihood estimation

2.5.2 Adaptive Bayes estimation

2.6 Example of real data estimation for gBm

2.7 Example of real data estimation for CIR

2.8 Hypotheses testing

2.9 AIC Model Selection

2.9.1 An example of AIC model selection for exchange rates data

2.10 LASSO model selection

2.10.1 An example of Lasso model selection for interest rates data

2.11 Change point estimation

2.11.1 Example of volatility change-point estimation for 2-dimensional SDE’s

2.11.2 An example of two stage estimation

2.11.3 Example of volatility change-point estimation in real data

2.12 Asynchronous covariance estimation

2.12.1 Other covariance estimators

2.13 Lead-lag estimation

2.13.1 Application of the lead-lag estimator to real data

2.14 Asymptotic expansion

2.14.1 Asymptotic expansion for general stochastic processes

 

3 Compound Poisson processes

3.1 Inhomogenous Compound Poisson Process

3.1.1 Linear intensity function

3.1.2 The Weibull model

3.1.3 The exponentially decaying intensity model

3.1.4 Modulated and periodical intensity model

3.1.5 Frequency modulation model

3.2 Multidimensional Compound Poisson Processes

3.2.1 Multivariate Gaussian Jumps

3.2.2 User specified jump distribution

3.3 Estimation

3.3.1 Compound Poisson process with Gaussian jumps

3.3.2 NIG Compound Poisson process

3.3.3 Exponential jump Compound Poisson process

3.3.4 The Weibull Compound Poisson process

 

4 Stochastic differential equations driven by Lévy processes

4.1 Lévy processes

4.1.1 Infinitely divisible distributions

4.1.2 Infinite divisible distributions, Lévy processes, Lévy-Itô decomposition

4.2 Wiener process

4.3 Compound Poisson process

4.4 Gamma process and its variants

4.4.1 Gamma process

4.4.2 Variance gamma process

4.4.3 Bilateral gamma process

4.4.4 Simulation of gamma processes

4.5 Generalized tempered stable process, tempered a stable process, CGMY process, positive tempered stable process

4.6 Inverse Gaussian process

4.7 Increasing stable process

4.8 Subordination

4.8.1 Definition

4.8.2 Compound Poisson process by subordination

4.8.3 Subordination of a Wiener process with drift

4.8.4 Variance gamma process with drift

4.8.5 Normal inverse Gaussian process

4.8.6 Normal tempered stable process

4.9 Stable process

4.10 Generalized hyperbolic processes

4.10.1 Generalized inverse Gaussian distribution

4.10.2 Generalized inverse Gaussian process and generalized hyperbolic process

4.10.3 GH distributions

4.10.4 Subclasses of the GH distributions

4.11 Stochastic differential equation driven by Lévy processes and their simulation

4.11.1 Semimartingale

4.11.2 Stochastic differential equations

4.11.3 Compound Poisson driving processes

4.11.4 Driving processes of code type

4.12 Estimation

4.12.1 Estimation of Jump-diffusion processes

4.12.2 Estimation of exponential Lévy processes

4.12.3 Bessel function of the third kind

 

5 Stochastic differential equations driven by the fractional Brownian motion

5.1 Model specification

5.2 Simulation of the fractional Gaussian noise

5.2.1 Cholesky method

5.2.2 Wood and Chan method

5.3 Simulation of fractional stochastic differential equations

5.4 Parametric inference for the fOU

5.4.1 Estimation of the Hurst exponent and the diffusion coefficient via quadratic generalized variations

5.4.2 Estimation of the drift parameter

5.5 An example on climate change data

 

6 CARMA models

6.1 Lévy driven CARMA Models

6.2 CARMA model specification

6.2.1 The yuima.carma-class

6.3 CARMA(p,q) model estimation

6.4 Examples of Lévy driven CARMA(p,q) models

6.4.1 Compound Poisson CARMA(2,1) process

6.4.2 Variance Gamma CARMA(2,1) process

6.4.3 Normal Inverse Gaussian CARMA(2,1) process

6.5 Application to the VIX index

 

7 COGARCH models

7.1 General order (p;q) model

7.1.1 How to input a COGARCH(p;q) model in yuima

7.1.2 Stationarity conditions

7.2 Simulation schemes

7.3 Generalized Method of Moments Estimation

7.3.1 Moments matching step

7.3.2 Lévy distribution estimation

7.4 Quasi-Maximum Likelihood Estimation

7.5 Relationship between GARCH(1,1) and COGARCH(1,1)

7.6 Application to real data

Reference

Index

O autorze

Stefano M. Iacus, PhD, is full professor of statistics in the Department of Economics, Management and Quantitative Methods at the University of Milan. He has been a member of the R Core Team (1999-2014) for the development of the R statistical environment and is now a member of the R Foundation. His research interests include inference for stochastic processes, simulation, computational statistics, causal inference, text mining, and sentiment analysis.

Nakahiro Yoshida, PhD, is full professor at the Graduate School of Mathematical Sciences, University of Tokyo. He is working in theoretical statistics, probability theory, computational statistics, and financial data analysis. He was awarded the Japan Statistical Society Award in 2009 and the Analysis Prize from the Mathematical Society of Japan in 2006.

Blurb

The YUIMA package is the first comprehensive R framework based on S4 classes and methods which allows for the simulation of stochastic differential equations driven by Wiener process, Lévy processes or fractional Brownian motion, as well as CARMA processes. The package performs various central statistical analyses such as quasi maximum likelihood estimation, adaptive Bayes estimation, structural change point analysis, hypotheses testing, asynchronous covariance estimation, lead-lag estimation, LASSO model selection, and so on. YUIMA also supports stochastic numerical analysis by fast computation of the expected value of functionals of stochastic processes through automatic asymptotic expansion by means of the Malliavin calculus. All models can be multidimensional, multiparametric or non parametric.The book explains briefly the underlying theory for simulation and inference of several classes of stochastic processes and then presents both simulation experiments and applications to real data. Although these processes have been originally proposed in physics and more recently in finance, they are becoming popular also in biology due to the fact the time course experimental data are now available. The YUIMA  package, already available on CRAN, can be freely downloaded and this companion book will make the user able to start his or her analysis from the first page.

• Contains both theory and code with step-by-step examples and figures
• Uses YUIMA package to implement the latest techniques available in the literature of inference for stochastic processes
• Shows how to create the description of very abstract models in the same way they are described in theoretical papers but with an extremely easy interface

Stefano M. Iacus, PhD, is full professor of statistics the Department of Economics, Management and Quantitative Methods at the University of Milan. He has been a member of the R Core Team (1999-2014) for the development of the R statistical environment and now member of the R Foundation. His research interests include inference for stochastic processes, simulation, computational statistics, causal inference, text mining, and sentiment analysis. 

Nakahiro Yoshida, PhD, is a professor at the Graduate School of Mathematical Sciences, University of Tokyo. He is working in theoretical statistics, probability theory, computational statistics, and financial data analysis. He was awarded the Japan Statistical Society Award in 2009 and the Analysis Prize from the Mathematical Society of Japan in 2006.