, c Simo Särkkä and Arno Solin 2019. This copy is made available for
personal use only and must not be adapted, sold or re-distributed.
Applied Stochastic Differential Equations
Simo Särkkä and Arno Solin
Applied Stochastic Differential Equations has been
published by Cambridge University Press, in the
IMS Textbooks series. It can be purchased directly
from Cambridge University Press.
Please cite this book as:
Simo Särkkä and Arno Solin (2019). Applied
Stochastic Differential Equations. Cambridge
University Press.
This PDF was compiled:
Friday 3rd May, 2019
, c Simo Särkkä and Arno Solin 2019. This copy is made available for
personal use only and must not be adapted, sold or re-distributed.
Contents
Preface ix
1 Introduction 1
2 Some Background on Ordinary Differential Equations 4
2.1 What Is an Ordinary Differential Equation? 4
2.2 Solutions of Linear Time-Invariant Differential Equations 6
2.3 Solutions of General Linear Differential Equations 10
2.4 Fourier Transforms 11
2.5 Laplace Transforms 13
2.6 Numerical Solutions of Differential Equations 16
2.7 Picard–Lindelöf Theorem 19
2.8 Exercises 20
3 Pragmatic Introduction to Stochastic Differential Equations 23
3.1 Stochastic Processes in Physics, Engineering, and Other Fields 23
3.2 Differential Equations with Driving White Noise 33
3.3 Heuristic Solutions of Linear SDEs 36
3.4 Heuristic Solutions of Nonlinear SDEs 39
3.5 The Problem of Solution Existence and Uniqueness 40
3.6 Exercises 40
4 Itô Calculus and Stochastic Differential Equations 42
4.1 The Stochastic Integral of Itô 42
4.2 Itô Formula 46
4.3 Explicit Solutions to Linear SDEs 49
4.4 Finding Solutions to Nonlinear SDEs 52
4.5 Existence and Uniqueness of Solutions 54
4.6 Stratonovich Calculus 55
4.7 Exercises 56
v
, c Simo Särkkä and Arno Solin 2019. This copy is made available for
personal use only and must not be adapted, sold or re-distributed.
vi Contents
5 Probability Distributions and Statistics of SDEs 59
5.1 Martingale Properties and Generators of SDEs 59
5.2 Fokker–Planck–Kolmogorov Equation 61
5.3 Operator Formulation of the FPK Equation 65
5.4 Markov Properties and Transition Densities of SDEs 67
5.5 Means and Covariances of SDEs 69
5.6 Higher-Order Moments of SDEs 72
5.7 Exercises 73
6 Statistics of Linear Stochastic Differential Equations 77
6.1 Means, Covariances, and Transition Densities of Linear SDEs 77
6.2 Linear Time-Invariant SDEs 80
6.3 Matrix Fraction Decomposition 83
6.4 Covariance Functions of Linear SDEs 87
6.5 Steady-State Solutions of Linear SDEs 90
6.6 Fourier Analysis of LTI SDEs 92
6.7 Exercises 96
7 Useful Theorems and Formulas for SDEs 98
7.1 Lamperti Transform 98
7.2 Constructions of Brownian Motion and the Wiener Measure 100
7.3 Girsanov Theorem 104
7.4 Some Intuition on the Girsanov Theorem 111
7.5 Doob’s h-Transform 113
7.6 Path Integrals 116
7.7 Feynman–Kac Formula 118
7.8 Exercises 124
8 Numerical Simulation of SDEs 126
8.1 Taylor Series of ODEs 126
8.2 Itô–Taylor Series–Based Strong Approximations of SDEs 129
8.3 Weak Approximations of Itô–Taylor Series 137
8.4 Ordinary Runge–Kutta Methods 140
8.5 Strong Stochastic Runge–Kutta Methods 144
8.6 Weak Stochastic Runge–Kutta Methods 151
8.7 Stochastic Verlet Algorithm 155
8.8 Exact Algorithm 157
8.9 Exercises 161
9 Approximation of Nonlinear SDEs 165
9.1 Gaussian Assumed Density Approximations 165
9.2 Linearized Discretizations 174
9.3 Local Linearization Methods of Ozaki and Shoji 175
personal use only and must not be adapted, sold or re-distributed.
Applied Stochastic Differential Equations
Simo Särkkä and Arno Solin
Applied Stochastic Differential Equations has been
published by Cambridge University Press, in the
IMS Textbooks series. It can be purchased directly
from Cambridge University Press.
Please cite this book as:
Simo Särkkä and Arno Solin (2019). Applied
Stochastic Differential Equations. Cambridge
University Press.
This PDF was compiled:
Friday 3rd May, 2019
, c Simo Särkkä and Arno Solin 2019. This copy is made available for
personal use only and must not be adapted, sold or re-distributed.
Contents
Preface ix
1 Introduction 1
2 Some Background on Ordinary Differential Equations 4
2.1 What Is an Ordinary Differential Equation? 4
2.2 Solutions of Linear Time-Invariant Differential Equations 6
2.3 Solutions of General Linear Differential Equations 10
2.4 Fourier Transforms 11
2.5 Laplace Transforms 13
2.6 Numerical Solutions of Differential Equations 16
2.7 Picard–Lindelöf Theorem 19
2.8 Exercises 20
3 Pragmatic Introduction to Stochastic Differential Equations 23
3.1 Stochastic Processes in Physics, Engineering, and Other Fields 23
3.2 Differential Equations with Driving White Noise 33
3.3 Heuristic Solutions of Linear SDEs 36
3.4 Heuristic Solutions of Nonlinear SDEs 39
3.5 The Problem of Solution Existence and Uniqueness 40
3.6 Exercises 40
4 Itô Calculus and Stochastic Differential Equations 42
4.1 The Stochastic Integral of Itô 42
4.2 Itô Formula 46
4.3 Explicit Solutions to Linear SDEs 49
4.4 Finding Solutions to Nonlinear SDEs 52
4.5 Existence and Uniqueness of Solutions 54
4.6 Stratonovich Calculus 55
4.7 Exercises 56
v
, c Simo Särkkä and Arno Solin 2019. This copy is made available for
personal use only and must not be adapted, sold or re-distributed.
vi Contents
5 Probability Distributions and Statistics of SDEs 59
5.1 Martingale Properties and Generators of SDEs 59
5.2 Fokker–Planck–Kolmogorov Equation 61
5.3 Operator Formulation of the FPK Equation 65
5.4 Markov Properties and Transition Densities of SDEs 67
5.5 Means and Covariances of SDEs 69
5.6 Higher-Order Moments of SDEs 72
5.7 Exercises 73
6 Statistics of Linear Stochastic Differential Equations 77
6.1 Means, Covariances, and Transition Densities of Linear SDEs 77
6.2 Linear Time-Invariant SDEs 80
6.3 Matrix Fraction Decomposition 83
6.4 Covariance Functions of Linear SDEs 87
6.5 Steady-State Solutions of Linear SDEs 90
6.6 Fourier Analysis of LTI SDEs 92
6.7 Exercises 96
7 Useful Theorems and Formulas for SDEs 98
7.1 Lamperti Transform 98
7.2 Constructions of Brownian Motion and the Wiener Measure 100
7.3 Girsanov Theorem 104
7.4 Some Intuition on the Girsanov Theorem 111
7.5 Doob’s h-Transform 113
7.6 Path Integrals 116
7.7 Feynman–Kac Formula 118
7.8 Exercises 124
8 Numerical Simulation of SDEs 126
8.1 Taylor Series of ODEs 126
8.2 Itô–Taylor Series–Based Strong Approximations of SDEs 129
8.3 Weak Approximations of Itô–Taylor Series 137
8.4 Ordinary Runge–Kutta Methods 140
8.5 Strong Stochastic Runge–Kutta Methods 144
8.6 Weak Stochastic Runge–Kutta Methods 151
8.7 Stochastic Verlet Algorithm 155
8.8 Exact Algorithm 157
8.9 Exercises 161
9 Approximation of Nonlinear SDEs 165
9.1 Gaussian Assumed Density Approximations 165
9.2 Linearized Discretizations 174
9.3 Local Linearization Methods of Ozaki and Shoji 175
