非线性期望的理论、方法及意义

发布时间：2018-04-02 20:42

本文选题：非线性数学期望　切入点：非线性正态分布　出处：《中国科学:数学》2017年10期

【摘要】：本文是非线性期望理论进展的一个综述,首先给出非线性期望空间的基本定义,并通过非线性期望的表示定理和几个典型的非线性独立同分布(i.i.d.)的例子来说明为什么这个新框架可以广泛地用来分析和计算现实世界(高维)数据背后隐藏的概率和统计分布的不确定性;进而介绍次线性期望空间中两个最重要的统计分布—非线性正态分布和最大分布,以及相应的非线性大数定律和中心极限定理,是新领域的基础性和关键性的突破,其典型的应用就是对于现实的(高维)样本数据的非常简单而深刻的φ-max-mean算法.本文还介绍一个最重要的连续时间随机过程——非线性Brown运动及相关随机分析,包括随机积分、随机微分方程和非线性鞅理论.新的理论框架实质性地推广了Kolmogorov于1933年建立的、以概率测度为核心的概率论公理体系(?,F,P).其关键不同的是,其核心概念是(非线性)期望ê,期望为线性的特殊情形对应着概率论公理体系.正是这种非线性使人们能够对于现实世界中无处不在的概率模型本身的不确定性也能进行定量的分析和计算.从而实质性地放宽了概率统计理论中对于现实世界的随机数据的统计假设要求,本文也因而获得了实际样本数据的非线性分布的φ-max-mean算法,它是一种新的非线性Monté-Carlo算法.
[Abstract]:This paper is a review of the basic theory of nonlinear expectation, the paper defines the nonlinear expectation space, and through the nonlinear expectation representation theorem and several typical nonlinear independent identically distributed (i.i.d.) example to explain why this new framework can be widely used in the analysis and calculation of the real world (high dimensional) behind the data probability and the statistical distribution of uncertainty; then introduces sublinear expectation space statistics of two of the most important nonlinear distribution of normal distribution and maximum distribution, and the nonlinear law of large numbers and the corresponding central limit theorem is a new field of the basic and key breakthrough, its typical application is for real (high dimensional) very simple but profound phi -max-mean algorithm sample data. This paper also introduces a most important continuous time stochastic process, nonlinear Brown movement and phase Close the stochastic analysis, including stochastic integral, nonlinear stochastic differential equation and martingale theory. New substantive theoretical framework to promote the Kolmogorov established in 1933, with the probability of probability measure is the core of the axiomatic system (?, F, P). The key difference is that its core concept is the expectation (nonlinear) as expected, special cases of linear corresponds to the probability axiom system. It is this kind of nonlinear analysis and calculation can make people everywhere for the probability model in the real world of uncertainty can be quantitatively. Thus substantially put wide statistical hypothesis of random data in the real world of the theory of probability and statistics in the requirements. This paper will have a nonlinear distribution of diameter -max-mean algorithm of actual data, it is a new kind of nonlinear Mont -Carlo algorithm.

【作者单位】：山东大学数学学院;山东大学高等研究院;
【基金】：国家自然科学基金(批准号:L1624032,11526205和11626247) 国家外专局111研究计划中国科学院和国家自然科学基金委员会交叉学科战略发展研究资助项目
【分类号】：O211.67

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