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2018 | Buch

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Stochastic Evolution Systems

Linear Theory and Applications to Non-Linear Filtering

verfasst von: Prof. Dr. Boris L. Rozovsky, Prof. Dr. Sergey V. Lototsky

Verlag: Springer International Publishing

Buchreihe : Probability Theory and Stochastic Modelling

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This monograph, now in a thoroughly revised second edition, develops the theory of stochastic calculus in Hilbert spaces and applies the results to the study of generalized solutions of stochastic parabolic equations.

The emphasis lies on second-order stochastic parabolic equations and their connection to random dynamical systems. The authors further explore applications to the theory of optimal non-linear filtering, prediction, and smoothing of partially observed diffusion processes. The new edition now also includes a chapter on chaos expansion for linear stochastic evolution systems.

This book will appeal to anyone working in disciplines that require tools from stochastic analysis and PDEs, including pure mathematics, financial mathematics, engineering and physics.

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Inhaltsverzeichnis

Frontmatter

Chapter 1. Examples and Auxiliary Results

Abstract

The first of the four sections in this chapter presents examples of linear stochastic evolution systems (LSESs) arising in various applications. The following three sections collect a number of auxiliary results which are used systematically throughout the book. In particular, Sect. 1.5 surveys the theory of stochastic ordinary differential equations.

Boris L. Rozovsky, Sergey V. Lototsky

Chapter 2. Stochastic Integration in a Hilbert Space

Abstract

This chapter is about stochastic calculus for continuous martingales and local martingales in a Hilbert space. The topics include definitions and investigations of martingales, local martingales and a Wiener process in a Hilbert space, construction of stochastic integrals with respect to these processes, and a detailed proof of the Itô formula for the square of a norm of a continuous semimartingale.

Boris L. Rozovsky, Sergey V. Lototsky

Chapter 3. Linear Stochastic Evolution Systems in Hilbert Spaces

Abstract

Fix \(T\in \mathbb {R}_+\) and consider a stochastic basis \(\mathbb {F}=(\varOmega , \mathscr {F}, \{\mathscr {F}_t\}_{t\in [0,T]}, \mathbb {P})\) with the usual assumptions. Let \((\mathbb {X},\mathbb {H},\mathbb {X}')\) be a normal triple of separable Hilbert spaces with canonical bi-linear functional [⋅, ⋅], and let \(\mathbb {Y}\) be another separable Hilbert space. As before, ∥⋅∥ is the norm in \(\mathbb {H}\) and (⋅, ⋅) is the inner product in \(\mathbb {H}\); the subscripts, such as \(\|\cdot \|{ }_{\mathbb {X}}\), are used for all other Hilbert space.

Boris L. Rozovsky, Sergey V. Lototsky

Chapter 4. Itô’s Second-Order Parabolic Equations

Abstract

Let us fix \(T_{0},T\in \mathbb {R}_{+}\) with T ₀ ≤ T, and \(\mathrm {d},\mathrm {d}_1\in \mathbb {N}\). We also fix the stochastic basis \(\mathbb {F}=(\varOmega , \mathscr {F}, \{\mathscr {F}_{t}\}_{t\in [0,T]}, \mathbb {P})\) with the usual assumptions, and a standard Wiener process w on \(\mathbb {F}\) with values in \(\mathbb {R}^{\mathrm {d}_1}.\)

Boris L. Rozovsky, Sergey V. Lototsky

Chapter 5. Itô’s Partial Differential Equations and Diffusion Processes

Abstract

In this chapter we continue the study of the Cauchy problem for second-order parabolic Itô equations, this time concentrating on qualitative, rather than analytical, aspects of the problem. The main objective is to establish various connections between these equations and diffusion processes.

Boris L. Rozovsky, Sergey V. Lototsky

Chapter 6. Filtering, Interpolation and Extrapolation of Diffusion Processes

Abstract

Recall that the filtering problem for diffusion processes first appeared in Sect. 1.2.2, where we discussed the motivation and general setting. Accordingly, we now go directly to the mathematical formulation of the problem.

Boris L. Rozovsky, Sergey V. Lototsky

Chapter 7. Hypoellipticity of Itô’s Second Order Parabolic Equations

Abstract

Smoothness of solutions of deterministic parabolic equations increases as the smoothness assumptions on their coefficients increase. This is a typical feature of parabolic equations. Moreover, under wide assumptions, the smoothness of solutions for t > 0 depends only on the smoothness of coefficients and does not depend on the smoothness of the initial functions. This is important, for example, in the study of the fundamental solution of a parabolic equation, since we can consider this solution as a solution of the corresponding Cauchy problem where the initial function is the Dirac delta function. Hypoellipticity is a particular case of the growth of smoothness property mentioned above.

Boris L. Rozovsky, Sergey V. Lototsky

Chapter 8. Chaos Expansion for Linear Stochastic Evolution Systems

Abstract

Separation of variables is widely used to study evolution equations. For deterministic equations, there are two variables to separate: time and space; the result is often an orthogonal expansion of the solution in the eigenfunctions of the operator in the equation.

Boris L. Rozovsky, Sergey V. Lototsky

Backmatter

Titel: Stochastic Evolution Systems
verfasst von: Prof. Dr. Boris L. Rozovsky
Prof. Dr. Sergey V. Lototsky
Copyright-Jahr: 2018
Verlag: Springer International Publishing
Electronic ISBN: 978-3-319-94893-5
Print ISBN: 978-3-319-94892-8
DOI: https://doi.org/10.1007/978-3-319-94893-5

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