Stochastic Filtering Theory
This book is based on a seminar given at the University of California at Los Angeles in the Spring of 1975. The choice of topics reflects my interests at the time and the needs of the students taking the course. Initially the lectures were written up for publication in the Lecture Notes series. How ever, when I accepted Professor A. V. Balakrishnan's invitation to publish them in the Springer series on Applications of Mathematics it became necessary to alter the informal and often abridged style of the notes and to rewrite or expand much of the original manuscript so as to make the book as self-contained as possible. Even so, no attempt has been made to write a comprehensive treatise on filtering theory, and the book still follows the original plan of the lectures. While this book was in preparation, the two-volume English translation of the work by R. S. Liptser and A. N. Shiryaev has appeared in this series. The first volume and the present book have the same approach to the sub ject, viz. that of martingale theory. Liptser and Shiryaev go into greater detail in the discussion of statistical applications and also consider inter polation and extrapolation as well as filtering.
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Martingales and the Wiener Process
Continuous Local Martingales
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apply assume assumption belongs bounded called Chapter Clearly complete condition consider constant contains continuous local martingale converges d-dimensional defined definition denote depend derivatives easy equals equivalent EXAMPLE exists fact filtering finite fixed follows formula function Furthermore Gaussian process give given Hence Hilbert space holds implies increasing process independent inequality interval L2-martingale Lemma limit linear Markov process mean measurable Note o-field observation obtain obvious operator positive predictable problem PROOF proved random variables real-valued relation relative Remark replaced representation respect result right-continuous right-hand side satisfies seen separable sequence shown simple solution square-integrable stochastic differential equation stochastic integral stochastic process stopping Suppose term Theorem theory tion uniformly values variation verify Wiener martingale Wiener process write