Simulation and Inference for Stochastic Processes with YUIMA: A Comprehensive R Framework for SDEs a

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