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2001 | Buch | 2. Auflage

Statistics of Random Processes

I. General Theory

verfasst von: Robert S. Liptser, Albert N. Shiryaev

Verlag: Springer Berlin Heidelberg

Buchreihe : Stochastic Modelling and Applied Probability

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Über dieses Buch

At the end of 1960s and the beginning of 1970s, when the Russian version of this book was written, the 'general theory of random processes' did not operate widely with such notions as semimartingale, stochastic integral with respect to semimartingale, the ItO formula for semimartingales, etc. At that time in stochastic calculus (theory of martingales), the main object was the square integrable martingale. In a short time, this theory was applied to such areas as nonlinear filtering, optimal stochastic control, statistics for diffusion­ type processes. In the first edition of these volumes, the stochastic calculus, based on square integrable martingale theory, was presented in detail with the proof of the Doob-Meyer decomposition for submartingales and the description of a structure for stochastic integrals. In the first volume ('General Theory') these results were used for a presentation of further important facts such as the Girsanov theorem and its generalizations, theorems on the innovation pro­ cesses, structure of the densities (Radon-Nikodym derivatives) for absolutely continuous measures being distributions of diffusion and Itô-type processes, and existence theorems for weak and strong solutions of stochastic differential equations. All the results and facts mentioned above have played a key role in the derivation of 'general equations' for nonlinear filtering, prediction, and smoothing of random processes.

Inhaltsverzeichnis

Frontmatter
Introduction
Abstract
A considerable number of problems in the statistics of random processes are formulated within the following scheme.
Robert S. Liptser, Albert N. Shiryaev
1. Essentials of Probability Theory and Mathematical Statistics
Abstract
According to Kolmogorov’s axiomatics the primary object of probability theory is the probability space (Ω,F, P). Here (Ω, F) denotes measurable space, i.e., a set Ω consisting of elementary events ω, with a distinguished system F of its subsets (events), forming a σ-algebra, and P denotes a probability measure (probability) defined on sets in F.
Robert S. Liptser, Albert N. Shiryaev
2. Martingales and Related Processes: Discrete Time
Abstract
Let (Ω, F, P) be a probability space, and let.F 1F 2 ⊆ ... ⊆ F N F be a nondecreasing family of sub-σ-algebras of F.
Robert S. Liptser, Albert N. Shiryaev
3. Martingales and Related Processes: Continuous Time
Abstract
Let (Ω, F, P) be a probability space and let F = (F t ), t ≥ 0, be a nondecreasing family of sub-σ-algebras of F.
Robert S. Liptser, Albert N. Shiryaev
4. The Wiener Process, the Stochastic Integral over the Wiener Process, and Stochastic Differential Equations
Abstract
Let (Ω, F, P) be a probability space and β = (β t), t ≥ 0, be a Brownian motion process (in the sense of the definition given in Section 1.4). Denote \(F_t^\beta= \sigma \left\{ {\omega :{\beta _s}} \right.,s \leqslant \left. t \right\}\) Then, according to (1.30) and (1.31),(P-a.s)
$$M\left( {{\beta _t}|F_s^\beta } \right) = {\beta _s},t \geqslant s $$
(4.1)
$$M\left[ {{{\left( {{\beta _t} - {\beta _s}} \right)}^2}|F_s^\beta } \right] = t - s,t \geqslant s. $$
(4.2)
Robert S. Liptser, Albert N. Shiryaev
5. Square Integrable Martingales and Structure of the Functionals on a Wiener Process
Abstract
Let (Ω, F, P) be a complete probability space, and let F = (F t ),t ≥ 0, be a nondecreasing (right continuous) family of sub-σ-algebras of F, each of which is augmented by sets from F having zero P-probability.
Robert S. Liptser, Albert N. Shiryaev
6. Nonnegative Supermartingales and Martingales, and the Girsanov Theorem
Abstract
Let (Ω, F, P) be a complete probability space, and let (F t ), 0 ≤ t T, be a nondecreasing family of sub-σ-algebras of F, augmented by sets from F of probability zero. Let W = (W t , F t ) be a Wiener process and let γ = (γ t , F t ) be a random process with
$$P\left( {\int {_0^T} \gamma _s^2ds \infty } \right) = 1.$$
(6.1)
Robert S. Liptser, Albert N. Shiryaev
7. Absolute Continuity of Measures corresponding to the Itô Processes and Processes of the Diffusion Type
Abstract
Let (Ω,F, P) be a complete probability space, let F = (F t ),t ≥ 0, be a nondecreasing family of sub-σ-algebras, and let W = (W t , F t ), t ≥ 0, be a Wiener process.
Robert S. Liptser, Albert N. Shiryaev
8. General Equations of Optimal Nonlinear Filtering, Interpolation and Extrapolation of Partially Observable Random Processes
Abstract
Let (Ω, F, P) be a complete probability space, and let (F t ), 0 ≤ tT, be a nondecreasing family of right continuous σ-algebras of F augmented by sets from F of zero probability.
Robert S. Liptser, Albert N. Shiryaev
9. Optimal Filtering, Interpolation and Extrapolation of Markov Processes with a Countable Number of States
Abstract
The present chapter will be concerned with a pair of random processes (θ, ξ) = (θ t , ξ t ), 0 ≤ tT, where the unobservable component θ is a Markov process with a finite or countable number of states, and the observable process ξ permits the stochastic differential
$$d{\xi _t}\,{A_t}({\theta _t},\xi )dt\, + \,{B_t}(\xi )d{W_t},$$
(9.1)
where W t is a Wiener process.
Robert S. Liptser, Albert N. Shiryaev
10. Optimal Linear Nonstationary Filtering
Abstract
On the probability space (Ω, F, P) with a distinguished family of the σ-algebras (F t ), tT, we shall consider the two-dimensional Gaussian random process (θ t , F t ), 0 ≤ tT, satisfying the stochastic differential equations
$$d{\theta _t}\, = \,a(t){\theta _t}dt\, + \,b(t)d{W_1}(t)$$
(10.1)
$$d{\xi _t}\, = \,A(t){\theta _t}dt\, + \,B(t)d{W_2}(t),$$
(10.2)
where W 1 = (W 1(t), F t ) and W 2 = (W 2(t), F t ) are two independent Wiener processes and θ 0, ξ 0 are F 0-measurable.
Robert S. Liptser, Albert N. Shiryaev
Backmatter
Metadaten
Titel
Statistics of Random Processes
verfasst von
Robert S. Liptser
Albert N. Shiryaev
Copyright-Jahr
2001
Verlag
Springer Berlin Heidelberg
Electronic ISBN
978-3-662-13043-8
Print ISBN
978-3-642-08366-2
DOI
https://doi.org/10.1007/978-3-662-13043-8