Simulation and inference for stochastic differential equations: with R examples
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| Physical Description: | XVIII, 284 S. graph. Darst. |
| ISBN: | 9780387758381 0387758380 9780387758398 |
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| 100 | 1 | |a Iacus, Stefano Maria |e Verfasser |0 (DE-588)171920104 |4 aut | |
| 245 | 1 | 0 | |a Simulation and inference for stochastic differential equations |b with R examples |c Stefano M. Iacus |
| 264 | 1 | |a New York, NY |b Springer |c 2008 | |
| 300 | |a XVIII, 284 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a Springer series in statistics | |
| 650 | 4 | |a Équations différentielles stochastiques | |
| 650 | 4 | |a Stochastic differential equations | |
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Contents
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-Nikodým
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
Ito
processes
. 32
XII Contents
1.10 Diffusion
processes
. 33
1.10.1 Ergodicity. 35
1.10.2
Markovianìty
. 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
Ito
formula
. 38
1.11.1
Orders of differentials in the
Ito
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 A'it-Sahalia model
. 50
1.13.8
Double-well potential
. 51
1.13.9
The Jacobi diffusion process
. 51
1.13.10 Ahn
and
Gao
model or inverse of Fellers 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
Contents XIII
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
7 = 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
CIR
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
XIV Contents
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.I 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 functions
. 225
A.
7
Loops and vectorization
. 227
A.8 Environments
. 228
A.
9
Time series objects
. 229
A.10
R
Scripts
. 231
A.ll Miscellanea
. 232
Appendix B: The sde Package
. 233
BM
. 234
cpoint
. 235
DBridge
. 236
dcElerian
. 237
dcEuler
. 238
dcKossler
. 238
dcOzaki
. 239
dcShoji
. 240
dcSim
. 241
DW.T
. 243
EULERloglik
. 243
gnim
. 245
Contents
XV
HPloglik . 247
ksmooth . 248
linear,
mart,
ef.
250
rcBS. 251
rcCIR. 252
rcOU. 253
rsCIR. 254
rsOU. 255
sde.sim. 256
sdeAIC . 259
SIMloglik. 261
simple.ef. 262
simple.ef2 . 264
References
. 267
Index. 279 |
| adam_txt |
Contents
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-Nikodým
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
Ito
processes
. 32
XII Contents
1.10 Diffusion
processes
. 33
1.10.1 Ergodicity. 35
1.10.2
Markovianìty
. 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
Ito
formula
. 38
1.11.1
Orders of differentials in the
Ito
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 A'it-Sahalia model
. 50
1.13.8
Double-well potential
. 51
1.13.9
The Jacobi diffusion process
. 51
1.13.10 Ahn
and
Gao
model or inverse of Fellers 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
Contents XIII
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
7 = 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
CIR
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
XIV Contents
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.I 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 functions
. 225
A.
7
Loops and vectorization
. 227
A.8 Environments
. 228
A.
9
Time series objects
. 229
A.10
R
Scripts
. 231
A.ll Miscellanea
. 232
Appendix B: The sde Package
. 233
BM
. 234
cpoint
. 235
DBridge
. 236
dcElerian
. 237
dcEuler
. 238
dcKossler
. 238
dcOzaki
. 239
dcShoji
. 240
dcSim
. 241
DW.T
. 243
EULERloglik
. 243
gnim
. 245
Contents
XV
HPloglik . 247
ksmooth . 248
linear,
mart,
ef.
250
rcBS. 251
rcCIR. 252
rcOU. 253
rsCIR. 254
rsOU. 255
sde.sim. 256
sdeAIC . 259
SIMloglik. 261
simple.ef. 262
simple.ef2 . 264
References
. 267
Index. 279 |
| any_adam_object | 1 |
| any_adam_object_boolean | 1 |
| author | Iacus, Stefano Maria |
| author_GND | (DE-588)171920104 |
| author_facet | Iacus, Stefano Maria |
| author_role | aut |
| author_sort | Iacus, Stefano Maria |
| author_variant | s m i sm smi |
| building | Verbundindex |
| bvnumber | BV023098561 |
| callnumber-first | Q - Science |
| callnumber-label | QA274 |
| callnumber-raw | QA274.23 |
| callnumber-search | QA274.23 |
| callnumber-sort | QA 3274.23 |
| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 820 SK 835 ST 250 |
| classification_tum | MAT 606f |
| ctrlnum | (OCoLC)191760004 (DE-599)DNB986188360 |
| dewey-full | 519.2 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 519 - Probabilities and applied mathematics |
| dewey-raw | 519.2 |
| dewey-search | 519.2 |
| dewey-sort | 3519.2 |
| dewey-tens | 510 - Mathematics |
| discipline | Informatik Mathematik |
| discipline_str_mv | Informatik Mathematik |
| format | Book |
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| id | DE-604.BV023098561 |
| illustrated | Illustrated |
| index_date | 2024-07-02T19:43:53Z |
| indexdate | 2024-07-20T09:31:07Z |
| institution | BVB |
| isbn | 9780387758381 0387758380 9780387758398 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-016301325 |
| oclc_num | 191760004 |
| open_access_boolean | |
| owner | DE-824 DE-91G DE-BY-TUM DE-945 DE-83 DE-11 DE-355 DE-BY-UBR DE-384 |
| owner_facet | DE-824 DE-91G DE-BY-TUM DE-945 DE-83 DE-11 DE-355 DE-BY-UBR DE-384 |
| physical | XVIII, 284 S. graph. Darst. |
| publishDate | 2008 |
| publishDateSearch | 2008 |
| publishDateSort | 2008 |
| publisher | Springer |
| record_format | marc |
| series2 | Springer series in statistics |
| spelling | Iacus, Stefano Maria Verfasser (DE-588)171920104 aut Simulation and inference for stochastic differential equations with R examples Stefano M. Iacus New York, NY Springer 2008 XVIII, 284 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Springer series in statistics Équations différentielles stochastiques Stochastic differential equations R Programm (DE-588)4705956-4 gnd rswk-swf Stochastische Differentialgleichung (DE-588)4057621-8 gnd rswk-swf Stochastische Differentialgleichung (DE-588)4057621-8 s R Programm (DE-588)4705956-4 s b DE-604 text/html http://deposit.dnb.de/cgi-bin/dokserv?id=3018756&prov=M&dok_var=1&dok_ext=htm Inhaltstext Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016301325&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Iacus, Stefano Maria Simulation and inference for stochastic differential equations with R examples Équations différentielles stochastiques Stochastic differential equations R Programm (DE-588)4705956-4 gnd Stochastische Differentialgleichung (DE-588)4057621-8 gnd |
| subject_GND | (DE-588)4705956-4 (DE-588)4057621-8 |
| title | Simulation and inference for stochastic differential equations with R examples |
| title_auth | Simulation and inference for stochastic differential equations with R examples |
| title_exact_search | Simulation and inference for stochastic differential equations with R examples |
| title_exact_search_txtP | Simulation and inference for stochastic differential equations with R examples |
| title_full | Simulation and inference for stochastic differential equations with R examples Stefano M. Iacus |
| title_fullStr | Simulation and inference for stochastic differential equations with R examples Stefano M. Iacus |
| title_full_unstemmed | Simulation and inference for stochastic differential equations with R examples Stefano M. Iacus |
| title_short | Simulation and inference for stochastic differential equations |
| title_sort | simulation and inference for stochastic differential equations with r examples |
| title_sub | with R examples |
| topic | Équations différentielles stochastiques Stochastic differential equations R Programm (DE-588)4705956-4 gnd Stochastische Differentialgleichung (DE-588)4057621-8 gnd |
| topic_facet | Équations différentielles stochastiques Stochastic differential equations R Programm Stochastische Differentialgleichung |
| url | http://deposit.dnb.de/cgi-bin/dokserv?id=3018756&prov=M&dok_var=1&dok_ext=htm http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016301325&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT iacusstefanomaria simulationandinferenceforstochasticdifferentialequationswithrexamples |