Statistics of Random Processes: I. General Theory / Edition 2by Robert Liptser, B. Aries, Albert N. Shiryaev
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, shastic differential… See more details below
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, shastic differential equations, the absolute continuity of probability measures for diffusion and Ito processes, elements of shastic 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.
- Springer Berlin Heidelberg
- Publication date:
- Stochastic Modelling and Applied Probability Series, #5
- Edition description:
- 2nd rev. and exp. ed. 2001
- Product dimensions:
- 9.21(w) x 6.14(h) x 1.00(d)
Table of Contents
Essentials of probability theory and mathematical statistics.- Martingales and related processes.- Martingales and supermartingales: continuous time.- The Wiener process, the shastic integral over the Wiener process, and shastic differential equations.- Square integrable martingales, and structure of the functionals on a Wiener process.- Nonnegative supermartingales and martingales, and the Girsanovs theorem.- Absolute continuity of measures corresponding to the Ito processes and processes of diffusion type.- General equations of optimal nonlinear filtering, interpolation and extrapolation of partially observable random processes.- Optimal filtering, interpolation and extrapolation of Markov processes with countable number of states.- Optimal linear nonstationary filtering.
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