Set-Valued Stochastic Integrals and Applications

verfasst von: Michał Kisielewicz

Verlag: Springer International Publishing

Buchreihe : Springer Optimization and Its Applications

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

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Über dieses Buch

This book is among the first concise presentations of the set-valued stochastic integration theory as well as its natural applications, as well as the first to contain complex approach theory of set-valued stochastic integrals. Taking particular consideration of set-valued Itô , set-valued stochastic Lebesgue, and stochastic Aumann integrals, the volume is divided into nine parts. It begins with preliminaries of mathematical methods that are then applied in later chapters containing the main results and some of their applications, and contains many new problems. Methods applied in the book are mainly based on functional analysis, theory of probability processes, and theory of set-valued mappings.
The volume will appeal to students of mathematics, economics, and engineering, as well as to mathematics professionals interested in applications of the theory of set-valued stochastic integrals.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Preliminaries

Abstract

In this chapter we present a survey of concepts and results of the fields of set theory, topology, functional analysis, and theory of stochastic processes that are used in the book. The greater part of all results is stated without proofs which can be found in the standard monographs. It is assumed that the basic notions of measure and probability theories are known to the reader.

Michał Kisielewicz

Chapter 2. Multifunctions

Abstract

In this chapter the basing notions of theory of set-valued mappings are presented. In particular, continuity and measurability of such mappings are considered. Set-valued mappings can be defined as relations that do not possess right-hand uniqueness property. Such mappings are also called as multifunctions.

Michał Kisielewicz

Chapter 3. Decomposable Subsets of

Abstract

In this chapter selected properties of decomposable subsets of the space of all Bochner p-integrable (equivalence classes of) functions with values in a Banach space (X, |⋅|) are considered. Furthermore, some properties of conditional expectations of subsets of the space \({\mathbb {L}}^p(T,\mathcal {F},\mu ,X)\) and set-valued martingales are presented.

Michał Kisielewicz

Chapter 4. Aumann Stochastic Integrals

Abstract

In this chapter we present the definition and properties of Aumann stochastic integrals of set-valued stochastic processes \(F:\mathbb {R}^+\times \Omega \rightarrow \mathrm {Cl}(\mathbb {R}^d)\) and subsets of the space \(\mathbb {L}^p(\mathbb {R}^+\times \Omega ,\beta \otimes \mathcal {F},\mathbb {R}^d)\). We begin with the definition and properties of the Aumann integrals of subsets of the space \({\mathbb {L}}^p(T,\mathcal {F},\mu ,X)\), where (X, |⋅|) is a separable Banach space.

Michał Kisielewicz

Chapter 5. Itô Set-Valued Integrals

Abstract

In this chapter we present the definition and properties of Itô set-valued integrals of square integrable non-anticipative matrix-valued stochastic processes. We begin with the definition and properties of Itô set-valued functional integrals of subsets of the space \({\mathbb {L}}^2(\mathbb {R}^+\times \Omega ,\Sigma _{\mathbb {F}},\mathbb {R}^{d\times m})\).

Michał Kisielewicz

Chapter 6. Stochastic Differential Inclusions

Abstract

In this chapter properties of stochastic differential inclusions are considered. The results of this chapter extend some result presented in the author monograph on the case of stochastic differential inclusions.

Michał Kisielewicz

Chapter 7. Set-Valued Stochastic Equations and Inclusions

Abstract

In this chapter we present properties of set-valued stochastic differential equations and set-valued functional inclusions. The results of this chapter extend some result presented in monograph on the case of set-valued stochastic differential equations.

Michał Kisielewicz

Chapter 8. Stochastic Optimal Control Problems

Abstract

This chapter contains some optimal control problems for systems described by stochastic differential inclusions. The existence of optimal controls and optimal solutions for such systems is a consequence of the weak compactness in distribution of the set \(\,\mathcal {Z}^x_D(F,\mathcal {G})\) defined in Remark 6.3.1 of Chapter 6, by the set of all weak solutions to (equivalence classes to) \(\,SDI(F,\mathcal {G})\,\) satisfying the initial condition x ₀ = x with \(x\in D\subset \mathbb {R}^d\). We begin with an introductory remark dealing with stochastic optimal control problems for systems described by stochastic differential equations depending on stochastic control parameters.

Michał Kisielewicz

Chapter 9. Mathematical Finance Problems

Abstract

Some optimal control problems of Financial Mathematics are presented. In particular, selected problems of optimal pricing and optimal portfolios in a given financial market are considered.

Michał Kisielewicz

Backmatter

Titel: Set-Valued Stochastic Integrals and Applications
verfasst von: Michał Kisielewicz
Verlag: Springer International Publishing
Electronic ISBN: 978-3-030-40329-4
Print ISBN: 978-3-030-40328-7
DOI: https://doi.org/10.1007/978-3-030-40329-4