Nonlinear Expectations And Stochastic Calculus Under Uncertainty
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Nonlinear Expectations and Stochastic Calculus Under Uncertainty
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.
Nonlinear Expectations and Stochastic Calculus under Uncertainty
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.
Real Options, Ambiguity, Risk and Insurance
Financial engineering has become the focus of widespread media attention as a result of the worldwide financial crisis of recent years. This book is the second in a series dealing with financial engineering from Ajou University in Korea. The main objective of the series is to disseminate recent developments and important issues in financial engineering to graduate students and researchers, and to provide surveys or pedagogical exposition of important published papers in a broad perspective, as well as analyses of important financial news concerning financial engineering research, practices or regulations. Real Options, Ambiguity, Risk and Insurance, comprises 12 chapters and is divided into three parts. In Part I, five chapters deal with real options analysis, which addresses the issue of investment decisions in complex, innovative or risky projects. Part II presents three chapters on ambiguity. The notion of ambiguity is one of the major breakthroughs in the expected utility theory; ambiguity arises as uncertainties cannot be precisely described in the probability space. Part III consists of four chapters devoted to risk and insurance, and covers mutual insurance for non-traded risks, downside risk management, and credit risk in fixed income markets. This volume will be useful to both graduate students and researchers in understanding relatively new areas in economics and finance, as well as challenging aspects of mathematics.