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1980 | Buch

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Stochastic Filtering Theory

verfasst von: Gopinath Kallianpur

Verlag: Springer New York

Buchreihe : Stochastic Modelling and Applied Probability

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Über dieses Buch

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.

Inhaltsverzeichnis

Frontmatter

1. Stochastic Processes: Basic Concepts and Definitions

Abstract

In this chapter and the next, we state a number of important results which are necessary for the work of the later chapters. Some of them might not be explicitly referred to in the later work, but they all form essential links in the chain of reasoning. To present the proofs of all of these results here would require preparatory background material which would considerably increase both the size and scope of this book. We therefore adopt the following approach with the aim of making the development of the text as self-contained as possible. We omit the proofs of those theorems which are treated in detail in well-known standard textbooks, such as P. A. Meyer’s book, Probability and Potentials [41]. However, those proofs will be presented which are not available in existing books and are to be found scattered in the literature, or which discuss ideas specially relevant to our purpose.

Gopinath Kallianpur

2. Martingales and the Wiener Process

Abstract

In the following definition T is taken to be either R ₊ or [0, T].

Gopinath Kallianpur

3. Stochastic Integrals

Abstract

Let L denote the family of all real-valued functions Y _t(ω) defined on R ₊ × Ω which are measurable with respect to ℬ(R ₊) × A and have the following properties:

Y = (Y _t) is adapted to (G _t).

For each ω,the function t → Y _t(ω) is left-continuous.

Gopinath Kallianpur

4. The Ito Formula

Abstract

A process M _t = (M _t ¹, ... ,M _t ^d) taking values in R ^d is a martingale with respect to the increasing σ-field family (F _t) if (M _t ⁱ, F _t) is a martingale for each i = 1, ... ,d, or equivalently, if (θ,M _t,) is a real-valued martingale with respect to (F _t) for every θ, ∈ R ^d . Here we use (,) to denote inner product in R ^d. (M _t, F _t) with M ₀ = 0 (a.s.) is a d-dimensional, continuous L ²-martingale if for every θ ∈ R ^d, (θ,M _t ) is a continuous L ²-martingale with respect toF _t. It is then easy to verify the existence of a unique d × d-matrix—valued process A _t, = (A _t ^ij) with the following properties:

Each A _t ^ij is F _t-measurable.

A ₀ = 0 and A _t(ω) is continuous in t for almost all ω.

For θ ∈ R ^d, (A _t θ,θ) is the (continuous) increasing process associated with (θ,M _t).

Gopinath Kallianpur

5. Stochastic Differential Equations

Abstract

Denote by C _d = C([0,T],R ^d), d ≥ 1 the space of continuous functions on [0,T] taking values in R ^d. Let S be a complete, separable metric space and D = D([0,T],S), the space of right-continuous functions from [0,T] to S having left-hand limits Let ℬ^t(C ^d) be the minimal σ-field with respect to which the coordinate functions $f\left( s \right)\left( {0 \leqslant s \leqslant t,f \in C_d } \right) $ are measurable. The a-field ℬ(D) is similarly defined. We write ℬ(C _d) = ℬ_T(C _d) and ℬ(D) = ℬ_T(D).

Gopinath Kallianpur

6. Functionals of a Wiener Process

Abstract

The purpose of this chapter is to derive representations of square-integrable functionals on Wiener space. This is a topic of importance in the theory of nonlinear prediction and filtering. The three main results in the literature derive for a square-integrable functional of a Wiener process (see definition below)

(a)

An L ²-convergent expansion in terms of Hermite functionals—Cameron-Martin

(b)

An L ²-convergent expansion in terms of multiple Wiener integrals—Ito

(c)

An Ito stochastic integral representation.

Gopinath Kallianpur

7. Absolute Continuity of Measures and Radon-Nikodym Derivatives

Abstract

As before, let (Ω,ℒ,P) be a complete probability space. Throughout this chapter it is assumed that (ℱ_t) t ∈ R ₊ or [0,T] is an increasing right-continuous family of σ-fields such that ℱ ₀ contains all P-null sets.

Gopinath Kallianpur

8. The General Filtering Problem and the Stochastic Equation of the Optimal Filter (Part I)

Abstract

Before discussing the filtering problem, we prove a number of results in preparation for the martingale approach to the stochastic differential equation of the optimal filter which will be derived in the later sections of this chapter. Let us recall that (Ω,A,P) is a complete probability space and (ℱ _t) (t ∈ R ₊) is an increasing family of sub σ-fields of A, and that it will be assumed that all P-null sets belong to ℱ₀. The following processes are given on Ω: (S _t) called the signal or system process; (Z _t), the observation process; and (B _t), the noise process. All three are related by the model

$$Z_t = S_t + B_t $$

(?8.1.1?)

Gopinath Kallianpur

9. Gaussian Solutions of Stochastic Equations

Abstract

Gaussian processes play an important role in the theory of linear filtering to be discussed in the next chapter. In the general stochastic filtering model it has been seen that the observation process and the innovation (Wiener) process are connected by an equation of the kind studied in Chapter 8. When the observation process is Gaussian, we have an example of the equation which will now be considered. The theory of stochastic equations whose solutions are Gaussian processes is an instructive special case of the general theory of functional stochastic differential equations because it is subsumed in the theory of nonanticipative representations of equivalent Gaussian measures and is identical with the latter if one of the measures is Wiener measure. We shall therefore present it in more detail than is strictly necessary for the purpose of solving linear filtering problems.

Gopinath Kallianpur

10. Linear Filtering Theory

Abstract

The most valuable achievements to date of the filtering theory of Chapter 8 belong to the linear theory which forms the subject of the present chapter and which is associated with the names of Kalman and Bucy. The Kalman filter (as this theory has come to be known) has a central place in our discussion of linear filtering not merely because of the fact that it is the precursor of the general nonlinear theory treated in Chapter 8 (and is still its most important special case) but because of its extensive applications in the post-Sputnik era to problems of tracking of satellites, signal detection, stochastic control, and aerospace engineering.

Gopinath Kallianpur

11. The Stochastic Equation of the Optimal Filter (Part II)

Abstract

In Theorems 8.4.3 and 8.4.4 of Section 8.4 the conditional expectation E ^t f(X _t) was shown to satisfy a stochastic differential equation for all f in D(Ã) for which the condition $\int_O^T {E\left| {f\left( {X_t } \right)h_t } \right|^2 < \infty } $ is fulfilled. However, the filtering problem can be regarded as completely solved if we can derive from (8.4.22) a stochastic differential equation for the conditional probability distribution—or the condition probability density—of X _t given, ℱ_t ^z and if, furthermore, it can be established that the equation has a unique solution. This was achieved in Chapter 10 for the linear theory, and we saw that the general equations of Chapter 8 yielded the Kalman filter. The complete solution of the optimal nonlinear filtering problem presents a much more difficult task.

Gopinath Kallianpur

Backmatter

Titel: Stochastic Filtering Theory
verfasst von: Gopinath Kallianpur
Verlag: Springer New York
Electronic ISBN: 978-1-4757-6592-2
Print ISBN: 978-1-4419-2810-8
DOI: https://doi.org/10.1007/978-1-4757-6592-2

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Über dieses Buch

Inhaltsverzeichnis

Frontmatter

1. Stochastic Processes: Basic Concepts and Definitions

2. Martingales and the Wiener Process

3. Stochastic Integrals

4. The Ito Formula

5. Stochastic Differential Equations

6. Functionals of a Wiener Process

7. Absolute Continuity of Measures and Radon-Nikodym Derivatives

8. The General Filtering Problem and the Stochastic Equation of the Optimal Filter (Part I)

9. Gaussian Solutions of Stochastic Equations

10. Linear Filtering Theory

11. The Stochastic Equation of the Optimal Filter (Part II)

Backmatter