Mathematics of Kalman-Bucy Filtering

1. Front Matter

   Pages I-X

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2. Elements of Probability Theory

   • Peter A. Ruymgaart, Tsu T. Soong

   Pages 1-29

3. Calculus in Mean Square

   • Peter A. Ruymgaart, Tsu T. Soong

   Pages 30-79

4. The Stochastic Dynamic System

   • Peter A. Ruymgaart, Tsu T. Soong

   Pages 80-99

5. The Kalman-Bucy Filter

   • Peter A. Ruymgaart, Tsu T. Soong

   Pages 100-153

6. A Theorem by Liptser and Shiryayev

   • Peter A. Ruymgaart, Tsu T. Soong

   Pages 154-157

7. Back Matter

   Pages 158-172

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Since their introduction in the mid 1950s, the filtering techniques developed by Kalman, and by Kalman and Bucy have been widely known and widely used in all areas of applied sciences. Starting with applications in aerospace engineering, their impact has been felt not only in all areas of engineering but also in the social sciences, biological sciences, medical sciences, as well as all other physical sciences. Despite all the good that has come out of this devel­ opment, however, there have been misuses because the theory has been used mainly as a tool or a procedure by many applied workers without them fully understanding its underlying mathematical workings. This book addresses a mathematical approach to Kalman-Bucy filtering and is an outgrowth of lectures given at our institutions since 1971 in a sequence of courses devoted to Kalman-Bucy filters. The material is meant to be a theoretical complement to courses dealing with applications and is designed for students who are well versed in the techniques of Kalman-Bucy filtering but who are also interested in the mathematics on which these may be based. The main topic addressed in this book is continuous-time Kalman-Bucy filtering. Although the discrete-time Kalman filter results were obtained first, the continuous-time results are important when dealing with systems developing in time continuously, which are hence more appropriately mod­ eled by differential equations than by difference equations. On the other hand, observations from the former can be obtained in a discrete fashion.

• Brownian motion
• Gaussian distribution
• Gaussian process
• Lévy process
• Mathematics
• Random variable
• equation
• filter
• filtering
• probability
• probability distribution
• probability space
• probability theory
• stochastic process
• stochastic processes

• Department of Mathematics, University of Technology, Delft, Delft, The Netherlands

  Peter A. Ruymgaart

• Faculty of Engineering and Applied Sciences, State University of New York at Buffalo, Buffalo, USA

  Tsu T. Soong

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