计算黎曼型积分的一种简单方法
A simple method of computing Riemann-type integrals
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中文总结 AI 辅助
本文提出一种基于代数伸缩恒等式与二次余项估计的统一方法，直接计算黎曼、黎曼-斯蒂尔杰斯、杨-孔杜拉尔、复解析及伊藤积分，并揭示伊藤修正与共形相消源于同一余项。
中文摘要 AI 辅助
标题中的“黎曼型积分”并非单一的新积分，而是经典构造族——黎曼、黎曼-斯蒂尔杰斯、杨-孔杜拉尔、复解析和伊藤积分——通过一种基本方法统一起来。我们给出一个简单论证，直接计算多项式的黎曼积分，从而建立多项式以及连续函数的微积分基本定理。同样的思想可推广到实直线、复平面、多维情形、黎曼-斯蒂尔杰斯积分，以及维纳过程（更一般地，连续半鞅）的多项式的伊藤积分，恢复经典伊藤公式。其贡献在于方法论：一个单一的代数伸缩恒等式，配以对二次余项的估计，驱动所有情形。变化的只是余项的归宿：对于 $\mathbb{R}$ 上的多项式，它恒为零；在 $\mathbb{C}$ 上，在二次变差条件 $Q(\pi)\to0$ 下以及对于赫尔德路径，它在极限中消失；而对于连续半鞅，它作为产生伊藤修正的二次变差存留下来——除了平面布朗运动，或更一般地，共形鞅，其中复平方内的代数相消再次将其移除。我们认为最后一点——即伊藤修正及其在共形情形下的相消是同一二次余项的两种实例，而非无关事实——是本文最具特色的观察。手稿分为三部分：第一部分在 $\mathbb{R}$ 和 $\mathbb{C}$ 上发展核心方法；第二部分将其确定性地推广到多维、黎曼-斯蒂尔杰斯和赫尔德（杨-孔杜拉尔）情形；第三部分将其推广到随机环境。
英文摘要
The "Riemann-type integrals" of the title are not a single new integral, but the family of classical constructions -- Riemann, Riemann-Stieltjes, Young-Kondurar, complex-analytic, and Itô -- unified by one elementary method. We give a simple argument computing the Riemann integral of a polynomial directly, establishing the fundamental theorem of calculus for polynomials and then continuous functions. The same idea extends to the real line, the complex plane, the multidimensional case, the Riemann-Stieltjes integral, and the Itô integral of a polynomial of a Wiener process (more generally, a continuous semimartingale), recovering the classical Itô formula. The contribution is methodological: a single algebraic telescoping identity, paired with an estimate on a quadratic remainder term, drives every case. What changes is only the remainder's fate: it vanishes identically for polynomials on $\mathbb{R}$; it vanishes in the limit under a quadratic-variation condition $Q(π)\to0$ on $\mathbb{C}$ and for Hölder paths; and for a continuous semimartingale it survives as the quadratic variation producing the Itô correction -- except for a planar Brownian motion, or more generally a conformal martingale, where an algebraic cancellation inside the complex square removes it again. We regard this last point -- that the Itô correction and its cancellation in the conformal case are two instances of the same quadratic remainder, not unrelated facts -- as the paper's most distinctive observation. The manuscript has three parts: Part I develops the core method on $\mathbb{R}$ and $\mathbb{C}$; Part II extends it deterministically to the multidimensional, Riemann-Stieltjes, and Hölder (Young-Kondurar) cases; Part III extends it to the stochastic setting.
