Venture into the uncharted territories of probability with a groundbreaking exploration of stochastic calculus under model uncertainty, guided by the pioneering work of Shige Peng. This book unveils the profound implications of sublinear expectations, offering a revolutionary approach to financial modeling, risk management, and stochastic analysis where classical probabilistic assumptions falter. Discover how sublinear expectation spaces provide a robust framework for handling ambiguity and Knightian uncertainty, surpassing the limitations of traditional linear expectations. Delve into the heart of G-Brownian motion, a cornerstone of this theory, and witness its unique characteristics unfold, challenging conventional notions of Brownian motion. Unravel the intricacies of Itô's integral in the G-setting, extending its power to a broader class of stochastic processes and paving the way for innovative applications in areas such as derivative pricing and hedging under uncertainty. The book meticulously constructs the mathematical machinery required to navigate this new landscape, from the representation theorem for sublinear expectations to the completion of functional spaces, ensuring a rigorous and accessible treatment of the subject. Explore the concept of capacity related to G-Expectation, essential for understanding the behavior of G-Brownian motion. Witness the extension of Itô's integral to stopping times, further enhancing the applicability of the theory. Master the intricacies of Itô's formula in the context of G-Brownian motion, a vital tool for solving stochastic differential equations and analyzing complex systems. This book is not merely a theoretical treatise; it is a practical guide for researchers, practitioners, and graduate students seeking to push the boundaries of stochastic analysis and unlock the potential of sublinear expectations in a world of ever-increasing uncertainty. Keywords: sublinear expectation, G-Brownian motion, Itô's integral, G-normal distribution, stochastic calculus, probability model uncertainty, capacity, stopping times, Itô's formula. Prepare to challenge your assumptions and redefine your understanding of probability in the face of uncertainty. This exploration of stochastic calculus transcends traditional boundaries, providing a robust framework for navigating the complexities of modern finance and beyond. Unearth the potential of G-Brownian motion, delve into the intricacies of Itô's integral, and master the art of stochastic analysis in a world where uncertainty reigns supreme, bridging the gap between theoretical concepts and practical applications in finance and risk management, offering a powerful toolkit for tackling real-world problems where traditional probabilistic models fall short.
Inhaltsverzeichnis (Table of Contents)
- Introduction
- Sublinear expectations and sublinear expectation spaces
- Preliminaries
- Representation of sublinear expectation
- Distributions and independence
- Completion of functional space and extension of sublinear expectation
- G-normal distribution
- G-Brownian motion and its characterization
- G-Expectation as an upper expectation
- Capacity related to G-Expectation
- Itô's integral with respect to G-Brownian motion
- Itô's integral with stopping times
- Quadratic variation of G-Brownian motion
- Distribution of <B>
- Itô's formula
- References
Zielsetzung und Themenschwerpunkte (Objectives and Key Themes)
This work aims to explore stochastic calculus within the framework of sublinear expectations, building upon the foundational work of Shige Peng. It delves into the properties of sublinear expectation spaces, G-normal distributions, and the construction of Itô's integral with respect to G-Brownian motion, extending the existing theory to a broader class of stochastic processes.
- Sublinear expectations and their representation.
- Properties and characterization of G-Brownian motion.
- Construction and properties of Itô's integral in the G-setting.
- Extension of Itô's integral to stopping times.
- Proof and application of Itô's formula in the context of G-Brownian motion.
Zusammenfassung der Kapitel (Chapter Summaries)
Introduction: This introduction provides a concise overview of the research on stochastic calculus under probability model uncertainty, pioneered by Shige Peng. It highlights the use of sublinear expectations to represent this uncertainty and briefly outlines the key concepts and results explored in the subsequent chapters, including the construction of G-Brownian motion and Itô's integral within this framework, emphasizing the extension beyond quasi-continuous processes.
Sublinear expectations and sublinear expectation spaces: This chapter introduces the fundamental concepts of sublinear expectations and their associated spaces, drawing upon the works of Peng, Hu, and others. It presents the representation theorem for sublinear expectations as suprema of linear expectations, a crucial result for understanding the nature of uncertainty in this setting. The chapter also defines distributions and independence within the sublinear expectation framework and establishes a norm to complete the sublinear expectation space, resulting in a Banach space. The chapter lays a rigorous mathematical foundation for the subsequent development of stochastic calculus in this non-linear setting, including careful definitions of monotonicity, sub-additivity, positive homogeneity, and the concept of dominance among nonlinear expectations. Examples are provided to illustrate the practical interpretation of sublinear expectations, such as in a game with uncertain probabilities.
Schlüsselwörter (Keywords)
Sublinear expectation, G-Brownian motion, Itô's integral, G-normal distribution, stochastic calculus, probability model uncertainty, capacity, stopping times, Itô's formula.
Häufig gestellte Fragen
Was ist das Ziel des Textes?
Das Ziel des Textes ist es, die stochastische Analysis im Rahmen sublinearer Erwartungen zu untersuchen, aufbauend auf den Arbeiten von Shige Peng. Er befasst sich mit den Eigenschaften sublinearer Erwartungsräume, G-Normalverteilungen und der Konstruktion des Itô-Integrals bezüglich der G-Brownschen Bewegung, wodurch die bestehende Theorie auf eine breitere Klasse stochastischer Prozesse erweitert wird.
Welche Hauptthemen werden behandelt?
Die Hauptthemen umfassen sublineare Erwartungen und ihre Darstellung, Eigenschaften und Charakterisierung der G-Brownschen Bewegung, Konstruktion und Eigenschaften des Itô-Integrals im G-Kontext, Erweiterung des Itô-Integrals auf Stoppzeiten sowie Beweis und Anwendung der Itô-Formel im Kontext der G-Brownschen Bewegung.
Was sind die Hauptkapitel und worum geht es darin?
Die Hauptkapitel sind:
- Einleitung: Bietet einen Überblick über die Forschung zur stochastischen Analysis unter Unsicherheit des Wahrscheinlichkeitsmodells.
- Sublineare Erwartungen und sublineare Erwartungsräume: Führt die Grundlagen sublinearer Erwartungen und zugehöriger Räume ein, einschließlich des Darstellungssatzes und Definitionen von Verteilungen und Unabhängigkeit.
Was ist die G-Normalverteilung?
Das Dokument befasst sich mit den Eigenschaften und der Charakterisierung der G-Brownschen Bewegung, die eng mit der G-Normalverteilung verbunden ist.
Was ist das Itô-Integral bezüglich der G-Brownschen Bewegung?
Ein wesentlicher Teil des Textes widmet sich der Konstruktion und den Eigenschaften des Itô-Integrals im G-Kontext.
Welche Schlüsselwörter werden verwendet?
Schlüsselwörter sind: Sublineare Erwartung, G-Brownsche Bewegung, Itô-Integral, G-Normalverteilung, stochastische Analysis, Unsicherheit des Wahrscheinlichkeitsmodells, Kapazität, Stoppzeiten, Itô-Formel.
Was wird in der Einführung behandelt?
Die Einleitung gibt einen Überblick über die Forschung zur stochastischen Analysis unter Unsicherheit des Wahrscheinlichkeitsmodells von Shige Peng und stellt die wichtigsten Konzepte und Ergebnisse vor, die in den folgenden Kapiteln untersucht werden.
Was behandelt das Kapitel "Sublineare Erwartungen und sublineare Erwartungsräume"?
Dieses Kapitel behandelt die grundlegenden Konzepte sublinearer Erwartungen, den Darstellungssatz, die Definitionen von Verteilungen und Unabhängigkeit innerhalb des sublinearen Erwartungsrahmens und die Vervollständigung des sublinearen Erwartungsraums.
- Citar trabajo
- Christian Bannasch (Autor), 2017, Stochastic Calculus Under Sublinear Expectation and Volatility Uncertainty, Múnich, GRIN Verlag, https://www.grin.com/document/512409
