Stochastic differential equations are differential equations whose solutions are stochastic processes. They exhibit appealing mathematical properties that are useful in modeling uncertainties and noisy phenomena in many disciplines. This book is motivated by applications of stochastic differential equations in target tracking and medical technology and, in particular, their use in methodologies such as filtering, smoothing, parameter estimation, and machine learning. It builds an intuitive hands-on understanding of what stochastic differential equations are all about, but also covers the essentials of Itô calculus, the central theorems in the field, and such approximation schemes as stochastic Runge–Kutta. Greater emphasis is given to solution methods than to analysis of theoretical properties of the equations. The book's practical approach assumes only prior understanding of ordinary differential equations. The numerous worked examples and end-of-chapter exercises include application-driven derivations and computational assignments. MATLAB/Octave source code is available for download, promoting hands-on work with the methods.
Contains worked examples and numerical simulation studies in each chapter which make ideas concrete
Includes downloadable MATLAB®/Octave source code to support application and adaptation
The gentle learning curve focuses on understanding and use rather than technical details
Author(s): Simo Särkkä and Arno Solin
Edition: 1
Publisher: Cambridge University Press
Year: 2019
Language: English
Pages: 324
Preface......Page 7
1 Introduction......Page 9
2.1 What Is an Ordinary Differential Equation?......Page 12
2.2 Solutions of Linear Time-Invariant Differential Equations......Page 14
2.3 Solutions of General Linear Differential Equations......Page 18
2.4 Fourier Transforms......Page 19
2.5 Laplace Transforms......Page 21
2.6 Numerical Solutions of Differential Equations......Page 24
2.7 Picard–Lindelöf Theorem......Page 27
2.8 Exercises......Page 28
3.1 Stochastic Processes in Physics, Engineering, and Other Fields......Page 31
3.2 Differential Equations with Driving White Noise......Page 41
3.3 Heuristic Solutions of Linear SDEs......Page 44
3.4 Heuristic Solutions of Nonlinear SDEs......Page 47
3.6 Exercises......Page 48
4.1 The Stochastic Integral of Itô......Page 50
4.2 Itô Formula......Page 54
4.3 Explicit Solutions to Linear SDEs......Page 57
4.4 Finding Solutions to Nonlinear SDEs......Page 60
4.5 Existence and Uniqueness of Solutions......Page 62
4.6 Stratonovich Calculus......Page 63
4.7 Exercises......Page 64
5.1 Martingale Properties and Generators of SDEs......Page 67
5.2 Fokker–Planck–Kolmogorov Equation......Page 69
5.3 Operator Formulation of the FPK Equation......Page 73
5.4 Markov Properties and Transition Densities of SDEs......Page 75
5.5 Means and Covariances of SDEs......Page 77
5.6 Higher-Order Moments of SDEs......Page 80
5.7 Exercises......Page 81
6.1 Means, Covariances, and Transition Densities of Linear SDEs......Page 85
6.2 Linear Time-Invariant SDEs......Page 88
6.3 Matrix Fraction Decomposition......Page 91
6.4 Covariance Functions of Linear SDEs......Page 95
6.5 Steady-State Solutions of Linear SDEs......Page 98
6.6 Fourier Analysis of LTI SDEs......Page 100
6.7 Exercises......Page 104
7.1 Lamperti Transform......Page 106
7.2 Constructions of Brownian Motion and the Wiener Measure......Page 108
7.3 Girsanov Theorem......Page 112
7.4 Some Intuition on the Girsanov Theorem......Page 119
7.5 Doob's h-Transform......Page 121
7.6 Path Integrals......Page 124
7.7 Feynman–Kac Formula......Page 126
7.8 Exercises......Page 132
8.1 Taylor Series of ODEs......Page 134
8.2 Itô–Taylor Series–Based Strong Approximations of SDEs......Page 137
8.3 Weak Approximations of Itô–Taylor Series......Page 145
8.4 Ordinary Runge–Kutta Methods......Page 148
8.5 Strong Stochastic Runge–Kutta Methods......Page 152
8.6 Weak Stochastic Runge–Kutta Methods......Page 159
8.7 Stochastic Verlet Algorithm......Page 163
8.8 Exact Algorithm......Page 165
8.9 Exercises......Page 169
9.1 Gaussian Assumed Density Approximations......Page 173
9.2 Linearized Discretizations......Page 182
9.3 Local Linearization Methods of Ozaki and Shoji......Page 183
9.4 Taylor Series Expansions of Moment Equations......Page 187
9.5 Hermite Expansions of Transition Densities......Page 191
9.6 Discretization of FPK......Page 193
9.7 Simulated Likelihood Methods......Page 200
9.8 Pathwise Series Expansions and the Wong–Zakai Theorem......Page 201
9.9 Exercises......Page 204
10 Filtering and Smoothing Theory......Page 205
10.1 Statistical Inference on SDEs......Page 206
10.2 Batch Trajectory Estimates......Page 211
10.3 Kushner–Stratonovich and Zakai Equations......Page 214
10.4 Linear and Extended Kalman–Bucy Filtering......Page 216
10.5 Continuous-Discrete Bayesian Filtering Equations......Page 219
10.6 Kalman Filtering......Page 224
10.7 Approximate Continuous-Discrete Filtering......Page 227
10.8 Smoothing in Continuous-Discrete and Continuous Time......Page 231
10.9 Approximate Smoothing Algorithms......Page 236
10.10 Exercises......Page 239
11.1 Overview of Parameter Estimation Methods......Page 242
11.2 Computational Methods for Parameter Estimation......Page 244
11.3 Parameter Estimation in Linear SDE Models......Page 247
11.4 Approximated-Likelihood Methods......Page 251
11.5 Likelihood Methods for Indirectly Observed SDEs......Page 254
11.6 Expectation–Maximization, Variational Bayes, and Other Methods......Page 256
11.7 Exercises......Page 257
12 Stochastic Differential Equations in Machine Learning......Page 259
12.1 Gaussian Processes......Page 260
12.2 Gaussian Process Regression......Page 262
12.3 Converting between Covariance Functions and SDEs......Page 265
12.4 GP Regression via Kalman Filtering and Smoothing......Page 273
12.5 Spatiotemporal Gaussian Process Models......Page 274
12.6 Gaussian Process Approximation of Drift Functions......Page 276
12.7 SDEs with Gaussian Process Inputs......Page 278
12.8 Gaussian Process Approximation of SDE Solutions......Page 280
12.9 Exercises......Page 282
13.1 Overview of the Covered Topics......Page 285
13.2 Choice of SDE Solution Method......Page 286
13.3 Beyond the Topics......Page 287
References......Page 289
Symbols and Abbreviations......Page 301
List of Examples......Page 313
List of Algorithms......Page 317
Index......Page 319
