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2019 | Buch

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Nonlinear Expectations and Stochastic Calculus under Uncertainty

with Robust CLT and G-Brownian Motion

verfasst von: Prof. Shige Peng

Verlag: Springer Berlin Heidelberg

Buchreihe : Probability Theory and Stochastic Modelling

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

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Über dieses Buch

This book is focused on the recent developments on problems of probability model uncertainty by using the notion of nonlinear expectations and, in particular, sublinear expectations. It provides a gentle coverage of the theory of nonlinear expectations and related stochastic analysis. Many notions and results, for example, G-normal distribution, G-Brownian motion, G-Martingale representation theorem, and related stochastic calculus are first introduced or obtained by the author.

This book is based on Shige Peng’s lecture notes for a series of lectures given at summer schools and universities worldwide. It starts with basic definitions of nonlinear expectations and their relation to coherent measures of risk, law of large numbers and central limit theorems under nonlinear expectations, and develops into stochastic integral and stochastic calculus under G-expectations. It ends with recent research topic on G-Martingale representation theorem and G-stochastic integral for locally integrable processes.

With exercises to practice at the end of each chapter, this book can be used as a graduate textbook for students in probability theory and mathematical finance. Each chapter also concludes with a section Notes and Comments, which gives history and further references on the material covered in that chapter.

Researchers and graduate students interested in probability theory and mathematical finance will find this book very useful.

Inhaltsverzeichnis

Frontmatter

Basic Theory of Nonlinear Expectations

Frontmatter

Chapter 1. Sublinear Expectations and Risk Measures

Abstract

A sublinear expectation is also called the upper expectation or the upper prevision, and this notion is used in situations when the probability models have uncertainty. In this chapter, we present the basic notion of sublinear expectations and the corresponding sublinear expectation spaces. We give the representation theorem of a sublinear expectation and the notions of distributions and independence within the framework of sublinear expectations. We also introduce a natural Banach norm of a sublinear expectation in order to get the completion of a sublinear expectation space which is a Banach space. As a fundamentally important example, we introduce the notion of coherent risk measures in finance.

Shige Peng

Chapter 2. Law of Large Numbers and Central Limit Theorem Under Probability Uncertainty

Abstract

In this chapter, we first introduce two types of fundamentally important distributions, namely, maximal distribution and a new type of nonlinear normal distribution—G-normal distribution in the theory of sublinear expectations. The former corresponds to constants and the latter corresponds to normal distribution in the classical probability theory. We then present the law of large numbers (LLN) and central limit theorem (CLT) under sublinear expectations. It is worth pointing out that the limit in LLN is a maximal distribution and the limit in CLT is a G-normal distribution.

Shige Peng

Stochastic Analysis Under G-Expectations

Frontmatter

Chapter 3. G-Brownian Motion and Itô’s Calculus

Abstract

The aim of this chapter is to introduce the concept of G-Brownian motion, study its properties and construct Itô’s integral with respect to G-Brownian motion. We emphasize here that this G-Brownian motion \(B_t\), \(t\ge 0\) is consistent with the classical one.

Shige Peng

Chapter 4. G-Martingales and Jensen’s Inequality

Abstract

In this chapter, we introduce the notion of G-martingales and the related Jensen’s inequality for a new type of G-convex functions. One essential difference from the classical situation is that here “M is a G-martingale” does not imply that “\(-M\) is a G-martingale”.

Shige Peng

Chapter 5. Stochastic Differential Equations

Abstract

In this chapter, we consider the stochastic differential equations and backward stochastic differential equations driven by G-Brownian motion. The conditions and proofs of existence and uniqueness of a stochastic differential equation is similar to the classical situation. However the corresponding problems for backward stochastic differential equations are not that easy, many are still open. We only give partial results to this direction.

Shige Peng

Chapter 6. Capacity and Quasi-surely Analysis for G-Brownian Paths

Abstract

In the last three chapters, we have considered random variables which are elements in a Banach space \(L^p_G(\Omega )\). A natural question is whether such elements \(\xi \) are still real functions defined on \(\Omega \), namely \(\xi =\xi (\omega ) \), \(\omega \in \Omega \).

Shige Peng

Stochastic Calculus for General Situations

Frontmatter

Chapter 7. G-Martingale Representation Theorem

Abstract

In Sect. 4.2 of Chap. 4, we presented a new type of G-martingale representation showing that a G-martingale can decomposed into a symmetric one and a decreasing one. Based on this idea, we now provide a complete and rigorous proof of this representation theorem for an \(L^2_G\)-martingale.

Shige Peng

Chapter 8. Some Further Results of Itô’s Calculus

Abstract

In this chapter, we use the quasi-surely analysis theory to develop Itô’s integrals without the quasi-continuity condition. This allows us to define Itô’s integral on stopping time interval. In particular, this new formulation can be applied to obtain Itô’s formula for a general \(C^{1,2}\)-function, thus extending previously available results.

Shige Peng

Backmatter

Titel: Nonlinear Expectations and Stochastic Calculus under Uncertainty
verfasst von: Prof. Shige Peng
Verlag: Springer Berlin Heidelberg
Electronic ISBN: 978-3-662-59903-7
Print ISBN: 978-3-662-59902-0
DOI: https://doi.org/10.1007/978-3-662-59903-7