Aimed primarily at those outside of the field of statistics, this book not only provides an accessible introduction to measure theory, stochastic calculus, and stochastic processes, with particular emphasis on martingales and Brownian motion, but develops into an excellent user's guide to filtering. Including exercises for students, it will be a complete resource for engineers, signal processing researchers or anyone with an interest in practical implementation of filtering techniques, in particular, the Kalman filter. Three separate chapters concentrate on applications arising in finance, genetics and population modelling.
Author(s): Lakhdar Aggoun, Robert J. Elliott
Series: Cambridge Series in Statistical and Probabilistic Mathematics
Publisher: Cambridge University Press
Year: 2004
Language: English
Pages: 270
Cover......Page 1
Half-title......Page 3
Series-title......Page 4
Title......Page 5
Copyright......Page 6
Contents......Page 7
Preface......Page 11
Part I Theory......Page 13
1.1 Random experiments and probabilities......Page 15
Probability measures......Page 18
1.2 Conditional probabilities and independence......Page 21
1.3 Random variables......Page 26
1.4 Conditional expectations......Page 40
1.5 Problems......Page 46
2.1 Definitions and general results......Page 50
2.2 Stopping times......Page 58
2.3 Discrete time martingales......Page 62
2.4 Doob decomposition......Page 68
2.5 Continuous time martingales......Page 71
2.6 Doob–Meyer decomposition......Page 74
2.7 Brownian motion......Page 82
Some properties of the Brownian motion process......Page 83
2.9 Brownian paths......Page 84
2.11 Problems......Page 87
Discrete-time processes......Page 91
Continuous-time processes......Page 96
3.3 Simple examples of stochastic integrals......Page 99
3.4 Stochastic integration with respect to a Brownian motion......Page 102
3.5 Stochastic integration with respect to general martingales......Page 106
3.6 The Itô formula for semimartingales......Page 109
3.7 The Itô formula for Brownian motion......Page 120
Representation results for Markov chains......Page 128
3.8 Representation results......Page 127
The single jump process......Page 132
3.9 Random measures......Page 135
Random measures associated with jump processes......Page 136
More of the differentiation rule......Page 138
3.10 Problems......Page 139
4.1 Introduction......Page 143
4.2 Measure change for discrete time processes......Page 146
A reverse measure change......Page 154
4.3 Girsanov’s Theorem......Page 157
4.4 The single jump process......Page 162
4.5 Change of parameter in Poisson processes......Page 169
4.6 Poisson process with drift......Page 173
4.7 Continuous-time Markov chains......Page 175
4.8 Problems......Page 177
Part II Applications......Page 179
5.3 Recursive estimation......Page 181
Recursive estimation......Page 187
5.5 The EM algorithm......Page 189
5.6 Discrete-time model parameter estimation......Page 190
Notation......Page 191
5.7 Finite-dimensional filters......Page 192
5.8 Continuous-time vector dynamics......Page 202
5.9 Continuous-time model parameters estimation......Page 208
Notation......Page 209
5.10 Direct parameter estimation......Page 218
The signal coefficient......Page 219
5.11 Continuous-time nonlinear filtering......Page 223
The correlated case......Page 225
5.12 Problems......Page 227
6.1 Volatility estimation......Page 229
Calibration......Page 231
Special cases......Page 232
6.2 Parameter estimation......Page 233
6.3 Filtering a price process......Page 234
6.4 Parameter estimation for a modified Kalman filter......Page 235
Parameter estimation......Page 238
6.5 Estimating the implicit interest rate of a risky asset......Page 241
Filtering......Page 242
Revising the parameters......Page 243
Numerical methods......Page 244
7.2 Recursive estimates......Page 247
7.3 Approximate formulae......Page 251
8.1 Introduction......Page 254
8.2 Distribution estimation......Page 255
8.3 Parameter estimation......Page 258
8.4 Pathwise estimation......Page 259
8.5 A Markov chain model......Page 260
8.7 A tags loss model......Page 262
8.8 Gaussian noise approximation......Page 265
References......Page 267
Index......Page 269
