Measure Theory and Filtering: Introduction and Applications (Cambridge Series in Statistical and Probabilistic Mathematics, Series Number 15)

This book was published in 2004. The estimation of noisily observed states from a sequence of data has traditionally incorporated ideas from Hilbert spaces and calculus-based probability theory. As conditional expectation is the key concept, the correct setting for filtering theory is that of a probability space. Graduate engineers, mathematicians and those working in quantitative finance wishing to use filtering techniques will find in the first half of this book an accessible introduction to measure theory, stochastic calculus, and stochastic processes, with particular emphasis on martingales and Brownian motion. Exercises are included. The book then provides an excellent users' guide to filtering: basic theory is followed by a thorough treatment of Kalman filtering, including recent results which extend the Kalman filter to provide parameter estimates. These ideas are then applied to problems arising in finance, genetics and population modelling in three separate chapters, making this a comprehensive resource for both practitioners and researchers. 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 Cover 1 Half-title 3 Series-title 4 Title 5 Copyright 6 Contents 7 Preface 11 Part I Theory 13 1 Basic probability concepts 15 1.1 Random experiments and probabilities 15 Probability measures 18 1.2 Conditional probabilities and independence 21 1.3 Random variables 26 1.4 Conditional expectations 40 1.5 Problems 46 2 Stochastic processes 50 2.1 Definitions and general results 50 2.2 Stopping times 58 2.3 Discrete time martingales 62 2.4 Doob decomposition 68 2.5 Continuous time martingales 71 2.6 Doob–Meyer decomposition 74 2.7 Brownian motion 82 Some properties of the Brownian motion process 83 2.8 Brownian motion process with drift 84 2.9 Brownian paths 84 2.10 Poisson process 87 2.11 Problems 87 3 Stochastic calculus 91 3.1 Introduction 91 3.2 Quadratic variations 91 Discrete-time processes 91 Continuous-time processes 96 3.3 Simple examples of stochastic integrals 99 3.4 Stochastic integration with respect to a Brownian motion 102 3.5 Stochastic integration with respect to general martingales 106 3.6 The Itô formula for semimartingales 109 3.7 The Itô formula for Brownian motion 120 Representation results for Markov chains 128 3.8 Representation results 127 The single jump process 132 3.9 Random measures 135 Random measures associated with jump processes 136 More of the differentiation rule 138 3.10 Problems 139 4 Change of measures 143 4.1 Introduction 143 4.2 Measure change for discrete time processes 146 A reverse measure change 154 4.3 Girsanov’s Theorem 157 4.4 The single jump process 162 4.5 Change of parameter in Poisson processes 169 4.6 Poisson process with drift 173 4.7 Continuous-time Markov chains 175 4.8 Problems 177 Part II Applications 179 5 Kalman filtering 181 5.1 Introduction 181 5.2 Discrete-time scalar dynamics 181 5.3 Recursive estimation 181 5.4 Vector dynamics 187 Recursive estimation 187 5.5 The EM algorithm 189 5.6 Discrete-time model parameter estimation 190 Notation 191 5.7 Finite-dimensional filters 192 5.8 Continuous-time vector dynamics 202 5.9 Continuous-time model parameters estimation 208 Notation 209 Estimation of B and D 218 5.10 Direct parameter estimation 218 The signal coefficient 219 The observation coefficient 223 5.11 Continuous-time nonlinear filtering 223 The correlated case 225 5.12 Problems 227 6 Financial applications 229 6.1 Volatility estimation 229 Calibration 231 Special cases 232 6.2 Parameter estimation 233 6.3 Filtering a price process 234 6.4 Parameter estimation for a modified Kalman filter 235 Kalman filter 238 Parameter estimation 238 6.5 Estimating the implicit interest rate of a risky asset 241 Filtering 242 Revising the parameters 243 Numerical methods 244 7 A genetics model 247 7.1 Introduction 247 7.2 Recursive estimates 247 7.3 Approximate formulae 251 8 Hidden populations 254 8.1 Introduction 254 8.2 Distribution estimation 255 8.3 Parameter estimation 258 Maximum posterior estimators 259 8.4 Pathwise estimation 259 Maximum posterior estimators 260 8.5 A Markov chain model 260 8.6 Recursive parameter estimation 262 8.7 A tags loss model 262 8.8 Gaussian noise approximation 265 References 267 Index 269 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. Aimed primarily at those outside of the field of statistics, this book not only provides both an accessible introduction to measure theory, stochastic calculus, and stochastic processes, with particular emphasis on martingales and Brownian motion, but develops into an excellent users' guide to filtering. Includes exercises for students. This is a complete resource for engineers, signal processing researchers or indeed anyone with an interest in practical implementation of filtering techniques, in particular the Kalman filter This book provides an accessible introduction to measure theory and stochastic calculus, and develops into an excellent users' guide to filtering. A complete resource for engineers, or anyone with an interest in implementation of filtering techniques. Three chapters concentrate on applications from finance, genetics and population modelling. Also includes exercises.