公理化方法重建布朗运动理论
首发时间:2020-02-25
摘要:布朗运动是一种具有连续时间参数和连续状态空间的随机过程,其理论不仅在概率论与随机过程中占有相当重要的地位,而且也是自然科学、工程技术和社会科学研究动态随机现象的重要数学工具。本文指出了维纳过程只能解释布朗运动随机变量统计特性、而不能描述布朗运动样本轨道运动规律的理论缺陷。此外,维纳过程为归纳总结出的数学定义,而不是依据概念和公理通过演绎推理方法得出的逻辑知识体系。本文使用公理化方法从空间和时间两个维度重建了布朗运动理论,推导出了布朗运动随机变量的统计特性,以及布朗运动样本轨道的导数、自相关函数、频域特性和位移公式,从而可全面、系统地阐明布朗运动现象、特征及规律。
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Theory of Brownian Motion reconstructed by axiomatic method
Abstract:Brownian motion is a random process with continuous time parameters and continuous state space.This paper points out that the Wiener process can only explain the statistical characteristics of random variables of Brownian motion, but can not describe the motion law of sample path of Brownian motion. In addition, the Wiener process is an inductive mathematical definition, not a logical system of knowledge derived from deductive reasoning based on concepts and axioms..The axiomatic method is used to reconstruct the Brownian motion theory from two dimensions of space and time.The statistical characteristics of the random variable of Brownian motion, as well as the derivative, autocorrelation function, frequency characteristics and displacement formula of the sample path of Brownian motion are derived.Therefore, the phenomenon, characteristics and laws of Brownian motion can be clarified comprehensively and systematically.
Keywords: stochastic processes Brownian motion Wiener process sample path
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