"Radically elementary " stochastic analysis for Wiener and Lévy processes via Internal Set Theory

随机分析是现代概率论中最繁荣的领域之一，在物理学和经济学中有着极其重要的应用，即使是在维纳空间上最基本的随机分析形式。然而，在作为随机分析基础的概念思想的美丽和简单与当前用于表述这些概念的数学理论的技术复杂性之间，存在着显著的对比。Nelson（1987）在其开创性的专著《根本初等概率论》（radical Elementary Probability Theory）中，基于内集理论（IST）的公理，开发了一种潜在的革命性的、根本简化的方法（通过无穷小）来研究连续时间随机过程的理论。IST，也是由于Nelson(1977)，是Robinson（1961）的非标准分析的公化，而Robinson（1961）的非标准分析又是一个一致的（相对于Zermelo- Fraenkel集合论），数学上严格的无限小分析框架。虽然罗宾逊（1961）对非标准分析的原始描述是基于数学逻辑的微妙结构，但纳尔逊（1987）的公理确实值得用“初等”这个谓词。我们将通过开发一个基本的随机分析框架来继续尼尔森的工作。首先，我们计划推导Itô公式、费曼-卡茨公式和格萨诺夫定理（在维纳空间上）的“极初等”版本。此后，我们计划从lsamvy过程开始，为更一般的跳跃扩散奠定基本随机演算的基础。

One of the most thriving areas of modern probability theory, with applications of paramount importance in physics and economics, is stochastic analysis, even in its most basic form of stochastic analysis on the Wiener space. There is a remarkable contrast, however, between the beauty and simplicity of the conceptual ideas underlying stochastic analysis and the technical sophistication of the current mathematical theory used to formulate these notions. In a pioneering monograph entitled Radically Elementary Probability Theory, Nelson (1987) has developed a potentially revolutionary, radically simplified approach (via infinitesimals) to the theory of continuous-time stochastic processes, based on the axioms of Internal Set Theory (IST). IST, also due to Nelson (1977), is an axiomatization of Robinson’s (1961) nonstandard analysis, which in turn is a consistent (relative to Zermelo- Fraenkel set theory), mathematically rigorous framework for analysis with infinitesimals. Whilst Robinson’s (1961) original account of nonstandard analysis is based on a delicate construction from mathematical logic, Nelson’s (1987) axioms indeed deserve the predicate “elementary”. We shall continue Nelson’s work through developing a radically elementary framework for stochastic analysis. First, we plan to derive “radically elementary” versions of Itô’s formula, of the Feynman-Kac formula and Girsanov’s theorem (on the Wiener space). Thereafter, we plan to lay the foundations of a radically elementary stochastic calculus with respect to more general jump-diffusions, starting with Lévy processes.