Applied Stochastic Differential Equations

1 C Simo S rkk and Arno Solin 2019. This copy is made available forpersonal use only and must not be adapted, sold or Stochastic Differential EquationsSimo S rkk and Arno SolinApplied Stochastic Differential Equationshas beenpublished by Cambridge University Press, in theIMS Textbooks series. It can be purchased directlyfrom Cambridge University cite this book as:Simo S rkk and Arno Solin (2019). AppliedStochastic Differential Equations . CambridgeUniversity PDF was compiled:Friday 3rdMay, 2019c Simo S rkk and Arno Solin 2019. This copy is made available forpersonal use only and must not be adapted, sold or Simo S rkk and Arno Solin 2019. This copy is made available forpersonal use only and must not be adapted, sold or Background on Ordinary Differential Is an Ordinary Differential Equation?

2 Of Linear Time-Invariant Differential of General Linear Differential Solutions of Differential Lindel f Introduction to Stochastic Differential Processes in Physics, Engineering, and Other Equations with Driving White Solutions of Linear Solutions of Nonlinear Problem of Solution Existence and Calculus and Stochastic Differential Stochastic Integral of It Solutions to Linear Solutions to Nonlinear and Uniqueness of Simo S rkk and Arno Solin 2019. This copy is made available forpersonal use only and must not be adapted, sold or Distributions and Statistics of Properties and Generators of Planck Kolmogorov Formulation of the FPK Properties and Transition Densities of and Covariances of Moments of of Linear Stochastic Differential , Covariances.

3 And Transition Densities of Linear Time-Invariant Fraction Functions of Linear Solutions of Linear Analysis of LTI Theorems and Formulas for of Brownian Motion and the Wiener Intuition on the Girsanov Kac Simulation of Series of Taylor Series Based Strong Approximations of Approximations of It Taylor Runge Kutta Stochastic Runge Kutta Stochastic Runge Kutta Verlet of Nonlinear Assumed Density Linearization Methods of Ozaki and Shoji175c Simo S rkk and Arno Solin 2019. This copy is made available forpersonal use only and must not be adapted, sold or Series Expansions of Moment Expansions of Transition of Likelihood Series Expansions and the Wong Zakai and Smoothing Inference on Trajectory Stratonovich and Zakai and Extended Kalman Bucy Bayesian Filtering Continuous-Discrete in Continuous-Discrete and Continuous Smoothing Exercises23111 Parameter Estimation in SDE of Parameter Estimation Methods for Parameter Estimation in Linear SDE Methods for Indirectly Observed Maximization, Variational Bayes.

4 And Differential Equations in Machine Process between Covariance Functions and Regression via Kalman Filtering and Gaussian Process Process Approximation of Drift with Gaussian Process Process Approximation of SDE of the Covered of SDE Solution Method278c Simo S rkk and Arno Solin 2019. This copy is made available forpersonal use only and must not be adapted, sold or the Topics279 References281 Symbols and Abbreviations293 List of Examples305 List of Algorithms309 Index311c Simo S rkk and Arno Solin 2019. This copy is made available forpersonal use only and must not be adapted, sold or book is an outgrowth of a set of lecture notes that has been extendedwith material from the doctoral theses of both authors and with a largeamount of completely new material.

5 The main motivation for the book isthe application of Stochastic Differential Equations (SDEs) in domains suchas target tracking and medical technology and, in particular, their use inmethodologies such as filtering, smoothing, parameter estimation, and ma-chine learning. We have also included a wide range of examples of appli-cations of SDEs arising in physics and electrical we are motivated by applications, much more emphasis is puton solution methods than on analysis of the theoretical properties of equa-tions. From the pedagogical point of view, one goal of this book is to pro-vide an intuitive hands-on understanding of what SDEs are all about, and ifthe reader wishes to learn the formal theory later, she can read, for example,the brilliant books of ksendal (2003) and Karatzas and Shreve (1991).

6 Another pedagogical aim is to overcome a slight disadvantage in manySDE books ( , the aforementioned ones), which is that they lean heavilyon measure theory, rigorous probability theory, and the theory of martin-gales. There is nothing wrong in these theories they are very powerfultheories and everyone should indeed master them. However, when thesetheories are explicitly used in explaining SDEs, they bring a flurry of tech-nical details that tend to obscure the basic ideas and intuition for the first-time reader. In this book, without shame, we trade rigor for readability bytreating SDEs completely without measure book s low learning curve only assumes prior knowledge of ordi-nary Differential Equations and basic concepts of statistics, together withunderstanding of linear algebra, vector calculus, and Bayesian book is mainly intended for advanced undergraduate and graduatestudents in Applied mathematics, signal processing, control engineering,ixc Simo S rkk and Arno Solin 2019.

7 This copy is made available forpersonal use only and must not be adapted, sold or , and computer science. However, the book is suitable also for re-searchers and practitioners who need a concise introduction to the topic ata level that enables them to implement or use the worked examples and numerical simulation studies in each chapterillustrate how the theory works in practice and can be implemented forsolving the problems. End-of-chapter exercises include application-drivenderivations and computational assignments. The MATLABR source codefor reproducing the example results is available for download through thebook s web page, promoting hands-on work with the have attempted to write the book to be freestanding in the sensethat it can be read without consulting other material on the way.

8 We havealso attempted to give pointers to work that either can be considered asthe original source of an idea or just contains more details on the topicat hand. However, this book is not a survey, but a textbook, and thereforewe have preferred citations that serve a pedagogical purpose, which mightnot always explicitly give credit to all or even the correct inventors of thetechnical ideas. Therefore, we need to apologize to any authors who havenot been cited although their work is clearly related to the topics that wecover. We hope you authors would like to thank Aalto University for providing thechance to write this book. We also would like to thank Robert Pich , PetteriPiiroinen, Roland Hostettler, Filip Tronarp, Santiago Cort s, Johan West ,Joonas Govenius, ngel Garc a-Fern ndez, Toni Karvonen, Juha Sarma-vuori, and Zheng Zhao for providing valuable comments on early versionsof the and Arnoc Simo S rkk and Arno Solin 2019.

9 This copy is made available forpersonal use only and must not be adapted, sold or topic of this book is Stochastic Differential Equations (SDEs). As theirname suggests, they really are Differential Equations that produce a differ-ent answer or solution trajectory each time they are solved. This peculiarbehaviour gives them properties that are useful in modeling of uncertain-ties in a wide range of applications, but at the same time it complicates therigorous mathematical treatment of emphasis of the book is on Applied rather than theoretical aspects ofSDEs and, therefore, we have chosen to structure the book in a way that webelieve supports learning SDEs from an Applied point of view.

10 In the fol-lowing, we briefly outline the purposes of each of the remaining chaptersand explain how the chapters are connected to each other. In the chapters,we have attempted to provide a wide selection of examples of the practicalapplication of theoretical and methodological results. Each chapter (exceptfor the Introduction and Epilogue) also contains a representative set of an-alytic and hands-on exercises that can be used for testing and deepeningunderstanding of the 2is a brief outline of concepts and solutions methods for deter-ministic ordinary Differential Equations (ODEs). We especially emphasizesolution methods for linear ODEs, because the methods translate quite eas-ily to SDEs.