Applied Stochastic Differential Equations 1st Edition by Simo Särkkä, Arno Solin 1316649466 9781316649466

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ISBN-10 :  1316649466 

ISBN-13 :  9781316649466

Author:  Simo Särkkä, Arno Solin 

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

Applied Stochastic Differential Equations 1st Table of contents:

1 Introduction
2 Some Background on Ordinary Differential Equations
2.1 What Is an Ordinary Differential Equation?
2.2 Solutions of Linear Time-Invariant Differential Equations
2.3 Solutions of General Linear Differential Equations
2.4 Fourier Transforms
2.5 Laplace Transforms
2.6 Numerical Solutions of Differential Equations
2.7 Picard–Lindelöf Theorem
2.8 Exercises
3 Pragmatic Introduction to Stochastic Differential Equations
3.1 Stochastic Processes in Physics, Engineering, and Other Fields
3.2 Differential Equations with Driving White Noise
3.3 Heuristic Solutions of Linear SDEs
3.4 Heuristic Solutions of Nonlinear SDEs
3.5 The Problem of Solution Existence and Uniqueness
3.6 Exercises
4 Itô Calculus and Stochastic Differential Equations
4.1 The Stochastic Integral of Itô
4.2 Itô Formula
4.3 Explicit Solutions to Linear SDEs
4.4 Finding Solutions to Nonlinear SDEs
4.5 Existence and Uniqueness of Solutions
4.6 Stratonovich Calculus
4.7 Exercises
5 Probability Distributions and Statistics of SDEs
5.1 Martingale Properties and Generators of SDEs
5.2 Fokker–Planck–Kolmogorov Equation
5.3 Operator Formulation of the FPK Equation
5.4 Markov Properties and Transition Densities of SDEs
5.5 Means and Covariances of SDEs
5.6 Higher-Order Moments of SDEs
5.7 Exercises
6 Statistics of Linear Stochastic Differential Equations
6.1 Means, Covariances, and Transition Densities of Linear SDEs
6.2 Linear Time-Invariant SDEs
6.3 Matrix Fraction Decomposition
6.4 Covariance Functions of Linear SDEs
6.5 Steady-State Solutions of Linear SDEs
6.6 Fourier Analysis of LTI SDEs
6.7 Exercises
7 Useful Theorems and Formulas for SDEs
7.1 Lamperti Transform
7.2 Constructions of Brownian Motion and the Wiener Measure
7.3 Girsanov Theorem
7.4 Some Intuition on the Girsanov Theorem
7.5 Doob’s h-Transform
7.6 Path Integrals
7.7 Feynman–Kac Formula
7.8 Exercises
8 Numerical Simulation of SDEs
8.1 Taylor Series of ODEs
8.2 Itô–Taylor Series–Based Strong Approximations of SDEs
8.3 Weak Approximations of Itô–Taylor Series
8.4 Ordinary Runge–Kutta Methods
8.5 Strong Stochastic Runge–Kutta Methods
8.6 Weak Stochastic Runge–Kutta Methods
8.7 Stochastic Verlet Algorithm
8.8 Exact Algorithm
8.9 Exercises
9 Approximation of Nonlinear SDEs
9.1 Gaussian Assumed Density Approximations
9.2 Linearized Discretizations
9.3 Local Linearization Methods of Ozaki and Shoji
9.4 Taylor Series Expansions of Moment Equations
9.5 Hermite Expansions of Transition Densities
9.6 Discretization of FPK
9.7 Simulated Likelihood Methods
9.8 Pathwise Series Expansions and the Wong–Zakai Theorem
9.9 Exercises
10 Filtering and Smoothing Theory
10.1 Statistical Inference on SDEs
10.2 Batch Trajectory Estimates
10.3 Kushner–Stratonovich and Zakai Equations
10.4 Linear and Extended Kalman–Bucy Filtering
10.5 Continuous-Discrete Bayesian Filtering Equations
10.6 Kalman Filtering
10.7 Approximate Continuous-Discrete Filtering
10.8 Smoothing in Continuous-Discrete and Continuous Time
10.9 Approximate Smoothing Algorithms
10.10 Exercises
11 Parameter Estimation in SDE Models
11.1 Overview of Parameter Estimation Methods
11.2 Computational Methods for Parameter Estimation
11.3 Parameter Estimation in Linear SDE Models
11.4 Approximated-Likelihood Methods
11.5 Likelihood Methods for Indirectly Observed SDEs
11.6 Expectation–Maximization, Variational Bayes, and Other Methods
11.7 Exercises
12 Stochastic Differential Equations in Machine Learning
12.1 Gaussian Processes
12.2 Gaussian Process Regression
12.3 Converting between Covariance Functions and SDEs
12.4 GP Regression via Kalman Filtering and Smoothing
12.5 Spatiotemporal Gaussian Process Models
12.6 Gaussian Process Approximation of Drift Functions
12.7 SDEs with Gaussian Process Inputs
12.8 Gaussian Process Approximation of SDE Solutions
12.9 Exercises
13 Epilogue
13.1 Overview of the Covered Topics
13.2 Choice of SDE Solution Method
13.3 Beyond the Topics

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Tags: Applied Stochastic, Differential Equations, Simo Särkkä, Arno Solin