Applied Stochastic Differential Equations: (Series Number 10 Institute of Mathematical Statistics Textbooks)

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.

Table of Contents:
1. Introduction; 2. Some background on ordinary differential equations; 3. Pragmatic introduction to stochastic differential equations; 4. Ito calculus and stochastic differential equations; 5. Probability distributions and statistics of SDEs; 6. Statistics of linear stochastic differential equations; 7. Useful theorems and formulas for SDEs; 8. Numerical simulation of SDEs; 9. Approximation of nonlinear SDEs; 10. Filtering and smoothing theory; 11. Parameter estimation in SDE models; 12. Stochastic differential equations in machine learning; 13. Epilogue.

About the Author :
Simo Särkkä is Associate Professor of Electrical Engineering and Automation at Aalto University, Finland, Technical Advisor at IndoorAtlas Ltd., and Adjunct Professor at Tampere University of Technology and Lappeenranta University of Technology. His research interests are in probabilistic modeling and sensor fusion for location sensing, health technology, and machine learning. He has authored over ninety peer-reviewed scientific articles as well as one book, titled Bayesian Filtering and Smoothing (Cambridge, 2013). Arno Solin is an Academy of Finland Postdoctoral Researcher with Aalto University, Finland and Technical Advisor at IndoorAtlas Ltd. His research interests focus on models and applications in sensor fusion for tracking and navigation, brain imaging, and machine learning problems. He has published over twenty peer-reviewed scientific papers, and has won several hackathons and competitions in mathematical modeling, including the 2014 Schizophrenia classification on Kaggle.

Review :
'Stochastic differential equations have long been used by physicists and engineers, especially in filtering and prediction theory, and more recently have found increasing application in the life sciences, finance and an ever-increasing range of fields. The authors provide intended users with an intuitive, readable introduction and overview without going into technical mathematical details from the often-demanding theory of stochastic analysis, yet clearly pointing out the pitfalls that may arise if its distinctive differences are disregarded. A large part of the book deals with underlying ideas and methods, such as analytical, approximative and computational, which are illustrated through many insightful examples. Linear systems, especially with additive noise and Gaussian solutions, are emphasized, though nonlinear systems are not neglected, and a large number of useful results and formulas are given. The latter part of the book provides an up to date survey and comparison of filtering and parameter estimation methods with many representative algorithms, and culminates with their application to machine learning.' Peter Kloeden, Johann Wolfgang Goethe-Universität Frankfurt am Main 'Overall, this is a very well-written and excellent introductory monograph to SDEs, covering all important analytical properties of SDEs, and giving an in-depth discussion of applied methods useful in solving various real-life problems.' Igor Cialenco, MathSciNet 'Chapters are rich in examples, numerical simulations, illustrations, derivations and computational assignment' Martin Ondreját, the European Mathematical Society and the Heidelberg Academy of Sciences and Humanities