Simulation and Inference for Stochastic Differential Equations (eBook)

Preface 7
Notation 16
Stochastic Processes and Stochastic Differential Equations 18
Elements of probability and random variables 18
Mean, variance, and moments 19
Change of measure and Radon-Nikodým derivative 21
Random number generation 22
The Monte Carlo method 22
Variance reduction techniques 25
Preferential sampling 26
Control variables 29
Antithetic sampling 30
Generalities of stochastic processes 31
Filtrations 31
Simple and quadratic variation of a process 32
Moments, covariance, and increments of stochastic processes 33
Conditional expectation 33
Martingales 35
Brownian motion 35
Brownian motion as the limit of a random walk 37
Brownian motion as L2[0,T] expansion 39
Brownian motion paths are nowhere differentiable 41
Geometric Brownian motion 41
Brownian bridge 44
Stochastic integrals and stochastic differential equations 46
Properties of the stochastic integral and Itô processes 49
Diffusion processes 50
Ergodicity 52
Markovianity 53
Quadratic variation 54
Infinitesimal generator of a diffusion process 54
How to obtain a martingale from a diffusion process 54
Itô formula 55
Orders of differentials in the Itô formula 55
Linear stochastic differential equations 56
Derivation of the SDE for the geometric Brownian motion 56
The Lamperti transform 57
Girsanov's theorem and likelihood ratio for diffusion processes 58
Some parametric families of stochastic processes 60
Ornstein-Uhlenbeck or Vasicek process 60
The Black-Scholes-Merton or geometric Brownian motion model 63
The Cox-Ingersoll-Ross model 64
The CKLS family of models 66
The modified CIR and hyperbolic processes 66
The hyperbolic processes 67
The nonlinear mean reversion Aït-Sahalia model 67
Double-well potential 68
The Jacobi diffusion process 68
Ahn and Gao model or inverse of Feller's square root model 69
Radial Ornstein-Uhlenbeck process 69
Pearson diffusions 69
Another classification of linear stochastic systems 71
One epidemic model 73
The stochastic cusp catastrophe model 74
Exponential families of diffusions 75
Generalized inverse gaussian diffusions 76
Numerical Methods for SDE 77
Euler approximation 78
A note on code vectorization 79
Milstein scheme 81
Relationship between Milstein and Euler schemes 82
Transform of the geometric Brownian motion 84
Transform of the Cox-Ingersoll-Ross process 84
Implementation of Euler and Milstein schemes: the sde.sim function 85
Example of use 86
The constant elasticity of variance process and strange paths 88
Predictor-corrector method 88
Strong convergence for Euler and Milstein schemes 90
KPS method of strong order =1.5 93
Second Milstein scheme 97
Drawing from the transition density 98
The Ornstein-Uhlenbeck or Vasicek process 99
The Black and Scholes process 99
The CIR process 99
Drawing from one model of the previous classes 100
Local linearization method 101
The Ozaki method 101
The Shoji-Ozaki method 103
Exact sampling 107
Simulation of diffusion bridges 114
The algorithm 115
Numerical considerations about the Euler scheme 117
Variance reduction techniques 118
Control variables 119
Summary of the function sde.sim 121
Tips and tricks on simulation 122
Parametric Estimation 124
Exact likelihood inference 127
The Ornstein-Uhlenbeck or Vasicek model 128
The Black and Scholes or geometric Brownian motion model 132
The Cox-Ingersoll-Ross model 134
Pseudo-likelihood methods 137
Euler method 137
Elerian method 140
Local linearization methods 142
Comparison of pseudo-likelihoods 143
Approximated likelihood methods 146
Kessler method 146
Simulated likelihood method 149
Hermite polynomials expansion of the likelihood 153
Bayesian estimation 170
Estimating functions 172
Simple estimating functions 172
Algorithm 1 for simple estimating functions 179
Algorithm 2 for simple estimating functions 182
Martingale estimating functions 187
Polynomial martingale estimating functions 188
Estimating functions based on eigenfunctions 193
Estimating functions based on transform functions 194
Discretization of continuous-time estimators 194
Generalized method of moments 197
The GMM algorithm 199
GMM, stochastic differential equations, and Euler method 200
What about multidimensional diffusion processes? 205
Miscellaneous Topics 206
Model identification via Akaike's information criterion 206
Nonparametric estimation 212
Stationary density estimation 213
Local-time and stationary density estimators 216
Estimation of diffusion and drift coefficients 217
Change-point estimation 223
Estimation of the change point with unknown drift 227
A famous example 230
Appendix A: A brief excursus into R 232
Typing into the R console 206
Assignments 212
R vectors and linear algebra 223
Subsetting 236
Different types of objects 237
Expressions and functions 240
Loops and vectorization 242
Environments 243
Time series objects 244
R Scripts 246
Miscellanea 247
Appendix B: The sde Package 248
BM 248
cpoint 250
DBridge 251
dcElerian 252
dcEuler 253
dcKessler 253
dcOzaki 254
dcShoji 255
dcSim 256
DWJ 257
EULERloglik 258
gmm 259
HPloglik 261
ksmooth 263
linear.mart.ef 264
rcBS 266
rcCIR 267
rcOU 268
rsCIR 269
rsOU 270
sde.sim 271
sdeAIC 273
SIMloglik 275
simple.ef 277
simple.ef2 278
References 281
Index 292