标题：Nonlinear expectation theory and stochastic calculus under Knightian uncertainty
作者机构：[Peng, S] School of Mathematics, Shandong University, Jinan, 250100, China
通讯作者地址：[Peng, S] School of Mathematics, Shandong UniversityChina
来源：Real Options, Ambiguity, Risk and Insurance: World Class University Program in Financial Engineering, Ajou University, Volume Two
关键词：Allais paradox; Ambiguity; Backward stochastic differential equation; Brownian motion; Ellsberg paradox; G-expectation; G-expectation; G-martingale; G-martingale; Itô integral and itô's calculus; Knightian uncertainty; Law of large numbers and central limit theory under uncertainty; Nonlinear expectation; Parabolic partial differential equation; Risk measure; Stochastic differential equation; Super-hedging; Uncertainty in economic theory; Vnmexpected utility theory
摘要：We review the developments in the theory of Backward Stochastic Differential Equations during the last 20 years, including the solutions' existence and uniqueness, comparison theorem, nonlinear Feynman-Kac formula, g-expectation and many other important results in BSDE theory and their applications to dynamic pricing and hedging in an incomplete financial market. We present our new framework of nonlinear expectation and its applications to financial risk measures under uncertainty of probability distributions. The generalized form of the law of large numbers and central limit theorem under sublinear expectation shows that the limit distribution is a sublinear G-normal distribution. A new type of Brownian motion, G-Brownian motion, is constructed which is a continuous stochastic process with independent and stationary increments under a sublinear expectation (or a nonlinear expectation). The corresponding robust version of Itô's calculus turns out to be a basic tool for problems of risk measures in finance and, more general, for decision theory under uncertainty. We also discuss a type of "fully nonlinear" BSDE under nonlinear expectation. © 2013 The authors and IOS Press. All rights reserved.