Nonlinear Expectations and Stochastic Calculus under Uncertainty: with Robust CLT and G-Brownian Motion

Description

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. Front Matter ....Pages i-xiii Front Matter ....Pages 1-1 Sublinear Expectations and Risk Measures (Shige Peng)....Pages 3-21 Law of Large Numbers and Central Limit Theorem Under Probability Uncertainty (Shige Peng)....Pages 23-45 Front Matter ....Pages 47-47 G-Brownian Motion and Itô’s Calculus (Shige Peng)....Pages 49-89 G-Martingales and Jensen’s Inequality (Shige Peng)....Pages 91-100 Stochastic Differential Equations (Shige Peng)....Pages 101-112 Capacity and Quasi-surely Analysis for G-Brownian Paths (Shige Peng)....Pages 113-143 Front Matter ....Pages 145-145 G-Martingale Representation Theorem (Shige Peng)....Pages 147-156 Some Further Results of Itô’s Calculus (Shige Peng)....Pages 157-170 Back Matter ....Pages 171-212