Preface ....................................................... VII
Notation ..................................................... XVII
1. Stochastic Processes and Stochastic Differential
Equations .................................................... 1
1.1. Elements of probability and random variables ........... 1
1.1.1. Mean, variance, and moments .................... 2
1.1.2. Change of measure and Radon-Nikodym
derivative ..................................... 4
1.2. Random number generation ............................... 5
1.3. The Monte Carlo method ................................. 5
1.4. Variance reduction techniques .......................... 8
1.4.1. Preferential sampling .......................... 9
1.4.2. Control variables ............................. 12
1.4.3. Antithetic sampling ........................... 13
1.5. Generalities of stochastic processes .................. 14
1.5.1. Filtrations ................................... 14
1.5.2. Simple and quadratic variation of a process ... 15
1.5.3. Moments, covariance. and increments of
stochastic processes .......................... 16
1.5.4. Conditional expectation ....................... 16
1.5.5. Martingales ................................... 18
1.6. Brownian motion ....................................... 18
1.6.1. Brownian motion as the limit of a
random walk ................................... 20
1.6.2. Brownian motion as L2[0,T] expansion .......... 22
1.6.3. Brownian motion paths are nowhere
differentiable ................................ 24
1.7. Geometric Brownian motion ............................. 24
1.8. Brownian bridge ....................................... 27
1.9. Stochastic integrals and stochastic differential
equations ............................................. 29
1.9.1. Properties of the stochastic integral and
Itô processes ................................. 32
1.10. Diffusion processes ................................... 33
1.10.1. Ergodicity .................................... 35
1.10.2. Markovianity .................................. 36
1.10.3. Quadratic variation ........................... 37
1.10.4. Infinitesimal generator of a diffusion
process ....................................... 37
1.10.5. How to obtain a martingale from a diffusion
process ....................................... 37
1.11. Itô formula ........................................... 38
1.11.1. Orders of differentials in the Itô formula .... 38
1.11.2. Linear stochastic differential equations ...... 39
1.11.3. Derivation of the SDE for the geometric
Brownian motion ............................... 39
1.11.4. The Lamperti transform ........................ 40
1.12. Girsanov's theorem and likelihood ratio for
diffusion processes ................................... 41
1.13. Some parametric families of stochastic processes ...... 43
1.13.1. Ornstein-Uhlenbeck or Vasicek process ......... 43
1.13.2. The Black-Scholes-Merton or geometric
Brownian motion model ......................... 46
1.13.3. The Cox-Ingersoll-Ross model .................. 47
1.13.4. The CKLS family of models ..................... 49
1.13.5. The modified CIR and hyperbolic processes ..... 49
1.13.6. The hyperbolic processes ...................... 50
1.13.7. The nonlinear mean reversion Aït-Sahalia
model ......................................... 50
1.13.8. Double-well potential ......................... 51
1.13.9. The Jacobi diffusion process .................. 51
1.13.10 Aim and Gao model or inverse of Feller's
square root model ............................. 52
1.13.11.Radial Ornstein-Uhlenbeck process ............. 52
1.13.12.Pearson diffusions ............................ 52
1.13.13.Another classification of linear
stochastic systems ............................ 54
1.13.14.One epidemic model ............................ 56
1.13.15.The stochastic cusp catastrophe model ......... 57
1.13.16.Exponential families of diffusions ............ 58
1.13.17.Generalized inverse gaussian diffusions ....... 59
2. Numerical Methods for SDE ................................... 61
2.1. Euler approximation ................................... 62
2.1.1. A note on code vectorization .................. 63
2.2. Milstein scheme ....................................... 65
2.3. Relationship between Milstein and Euler schemes ....... 66
2.3.1. Transform of the geometric Brownian motion .... 68
2.3.2. Transform of the Cox-Ingersoll-Ross process ... 68
2.4. Implementation of Euler and Milstein schemes:
the sde.sim function .................................. 69
2.4.1. Example of use ................................ 70
2.5. The constant elasticity of variance process
and strange paths ..................................... 72
2.6. Predictor-corrector method ............................ 72
2.7. Strong convergence for Euler and Milstein schemes ..... 74
2.8. KPS method of strong order у = 1.5 .................... 77
2.9. Second Milstein scheme ................................ 81
2.10. Drawing from the transition density ................... 82
2.10.1. The Ornstein-Uhlenbeck or Vasicek process ..... 83
2.10.2. The Black and Scholes process ................. 83
2.10.3. The CTR process ............................... 83
2.10.4. Drawing from one model of the previous
classes ....................................... 84
2.11. Local linearization method ............................ 85
2.11.1. The Ozaki method .............................. 85
2.11.2. The Shoji-Ozaki method ........................ 87
2.12 Exact sampling ........................................ 91
2.13 Simulation of diffusion bridges ....................... 98
2.13.1. The algorithm ................................. 99
2.14. Numerical considerations about the Euler scheme ...... 101
2.15. Variance reduction techniques ........................ 102
2.15.1 Control variables ............................. 103
2.16. Summary of the function sde.sim ...................... 105
2.17. Tips and tricks on simulation ........................ 106
3. Parametric Estimation ...................................... 109
3.1. Exact likelihood inference ........................... 112
3.1.1. The Ornstein-Uhlenbeck or Vasicek model ...... 113
3.1.2. The Black and Scholes or geometric
Brownian motion model ........................ 117
3.1.3. The Cox-Ingersoll-Ross model ................. 119
3.2. Pseudo-likelihood methods ............................ 122
3.2.1. Euler method ................................. 122
3.2.2. Elerian method ............................... 125
3.2.3. Local linearization methods .................. 127
3.2.4. Comparison of pseudo-likelihoods ............. 128
3.3 Approximated likelihood methods ...................... 131
3.3.1. Kessler method ............................... 131
3.3.2. Simulated likelihood method .................. 134
3.3.3. Hermite polynomials expansion of the
likelihood ................................... 138
3.4. Bayesian estimation .................................. 155
3.5. Estimating functions ................................. 157
3.5.1. Simple estimating functions .................. 157
3.5.2. Algorithm 1 for simple estimating
functions .................................... 164
3.5.3. Algorithm 2 for simple estimating
functions .................................... 167
3.5.4. Martingale estimating functions .............. 172
3.5.5. Polynomial martingale estimating
functions .................................... 173
3.5.6. Estimating functions based on
eigenfunctions ............................... 178
3.5.7. Estimating functions based on transform
functions .................................... 179
3.6. Discretization of continuous-time estimators ......... 179
3.7. Generalized method of moments ........................ 182
3.7.1. The GMM algorithm ............................ 184
3.7.2. GMM. stochastic differential equations,
and Euler method ............................. 185
3.8. What about multidimensional diffusion processes? ..... 190
4. Miscellaneous Topics ....................................... 191
4.1. Model identification via Akaike's information
criterion ............................................ 191
4.2. Nonparametric estimation ............................. 197
4.2.1. Stationary density estimation ................ 198
4.2.2. Local-time and stationary density
estimators ................................... 201
4.2.3. Estimation of diffusion and drift
coefficients ................................. 202
4.3. Change-point estimation .............................. 208
4.3.1. Estimation of the change point with
unknown drift ................................ 212
4.3.2. A famous example ............................. 215
Appendix A: A brief excursus into R ........................ 217
A.l. Typing into the R console ....................... 217
A.2. Assignments ..................................... 218
A.3. R vectors and linear algebra .................... 220
A.4. Subsetting ...................................... 221
A.5. Different types of objects ...................... 222
A.6. Expressions and fimctions ....................... 225
A.7. Loops and vectorization ......................... 227
A.8. Environments .................................... 228
A.9. Time series objects ............................. 229
A.10.R Scripts ....................................... 231
A.11.Miscellanea ..................................... 232
Appendix B: The sde Package ................................ 233
BM ................................................... 234
cpoint ............................................... 235
DBridge .............................................. 236
dcElerian ............................................ 237
dcEuler .............................................. 238
dcKessler ............................................ 238
dcOzaki .............................................. 239
dcShoji .............................................. 240
dcSim ................................................ 241
DWJ .................................................. 243
EULERloglik .......................................... 243
gmm .................................................. 245
HPloglik ............................................. 247
ksmooth .............................................. 248
linear.mart.ef ....................................... 250
rcBS ................................................. 251
rc-CIR ............................................... 252
rcOU ................................................. 253
rsCIR ................................................ 254
rsOU ................................................. 255
sde.sim .............................................. 256
sdcAIC ............................................... 259
SIMloglik ............................................ 261
simple.ef ............................................ 262
simple.ef2 ........................................... 264
References .................................................... 267
Index ......................................................... 279
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