Statistics of Random Processes: I. General Theory

✍ Scribed by Robert S. Liptser, Albert N. Shiryaev (auth.)

• Publisher: Springer-Verlag Berlin Heidelberg
• Year: 2001
• Tongue: English
• Leaves: 433
• Series: Applications of Mathematics 5
• Edition: 2
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

Front Matter....Pages I-XV
Introduction....Pages 1-10
Essentials of Probability Theory and Mathematical Statistics....Pages 11-37
Martingales and Related Processes: Discrete Time....Pages 39-56
Martingales and Related Processes: Continuous Time....Pages 57-84
The Wiener Process, the Stochastic Integral over the Wiener Process, and Stochastic Differential Equations....Pages 85-159
Square Integrable Martingales and Structure of the Functionals on a Wiener Process....Pages 161-218
Nonnegative Supermartingales and Martingales, and the Girsanov Theorem....Pages 219-249
Absolute Continuity of Measures corresponding to the Itô Processes and Processes of the Diffusion Type....Pages 251-315
General Equations of Optimal Nonlinear Filtering, Interpolation and Extrapolation of Partially Observable Random Processes....Pages 317-350
Optimal Filtering, Interpolation and Extrapolation of Markov Processes with a Countable Number of States....Pages 351-373
Optimal Linear Nonstationary Filtering....Pages 375-407
Back Matter....Pages 409-427

✦ Subjects

Probability Theory and Stochastic Processes; Statistical Theory and Methods