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AuthorLiptser, Robert S. author
TitleStatistics of Random Processes [electronic resource] : I. General Theory / by Robert S. Liptser, Albert N. Shiryaev
ImprintBerlin, Heidelberg : Springer Berlin Heidelberg : Imprint: Springer, 2001
Edition Second, Revised and Expanded Edition
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Descript XV, 427 p. online resource


The subject of these two volumes is non-linear filtering (prediction and smoothing) theory and its application to the problem of optimal estimation, control with incomplete data, information theory, and sequential testing of hypothesis. The required mathematical background is presented in the first volume: the theory of martingales, stochastic differential equations, the absolute continuity of probability measures for diffusion and Ito processes, elements of stochastic calculus for counting processes. The book is not only addressed to mathematicians but should also serve the interests of other scientists who apply probabilistic and statistical methods in their work. The theory of martingales presented in the book has an independent interest in connection with problems from financial mathematics. In the second edition, the authors have made numerous corrections, updating every chapter, adding two new subsections devoted to the Kalman filter under wrong initial conditions, as well as a new chapter devoted to asymptotically optimal filtering under diffusion approximation. Moreover, in each chapter a comment is added about the progress of recent years


1. Essentials of Probability Theory and Mathematical Statistics -- 2. Martingales and Related Processes: Discrete Time -- 3. Martingales and Related Processes: Continuous Time -- 4. The Wiener Process, the Stochastic Integral over the Wiener Process, and Stochastic Differential Equations -- 5. Square Integrable Martingales and Structure of the Functionals on a Wiener Process -- 6. Nonnegative Supermartingales and Martingales, and the Girsanov Theorem -- 7. Absolute Continuity of Measures corresponding to the Itรด Processes and Processes of the Diffusion Type -- 8. General Equations of Optimal Nonlinear Filtering, Interpolation and Extrapolation of Partially Observable Random Processes -- 9. Optimal Filtering, Interpolation and Extrapolation of Markov Processes with a Countable Number of States -- 10. Optimal Linear Nonstationary Filtering

Mathematics Dynamics Ergodic theory Probabilities Statistics Mathematics Dynamical Systems and Ergodic Theory Probability Theory and Stochastic Processes Statistical Theory and Methods


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