Nonlinear Expectations and Stochastic Calculus under Uncertainty
Nonlinear Expectations and Stochastic Calculus under Uncertainty
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DOI：
10.1007/978-3-662-59903-7
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发表时间：
2010-02
期刊：
影响因子：
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通讯作者：
S. Peng
中科院分区：
文献类型：
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作者：
S. Peng
本文回顾了近20年来倒向随机微分方程理论的发展，包括解的存在唯一性、比较定理、非线性费曼-卡茨公式、g-期望和其他许多重要结果，以及它们在不完全金融市场动态定价和套期保值中的应用。本文提出了新的非线性期望框架及其在概率分布不确定性下的金融风险度量中的应用。次线性期望下大数定律和中心极限定理的推广形式表明极限分布是次线性g -正态分布。构造了一种新的布朗运动——g -布朗运动，它是在次线性期望（或非线性期望）下具有独立平稳增量的连续随机过程。Itô的微积分对应的健壮版本被证明是解决金融风险度量问题的基本工具，更一般地说，是解决不确定性下的决策理论的基本工具。我们还讨论了一类在非线性期望下的“全非线性”BSDE。
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.