子集格的多面体与随机变量的最大期望
Polytopes associated with lattices of subsets and maximising expectation of random variables
摘要：
本论文源于金融数学中的一个问题，涉及计算基于多个资产的欧式看涨期权的价值，每个资产都遵循二项式模型。该模型导致了一个有趣的家庭的多面体$P(b)$与功率集$\mathcal{L} = \wp\{1,\dots,m\}$和参数化的$b \in \mathbb{R}^m$，其中每一个是一个收集的概率密度函数$\mathcal{L}$。对于每个非空的$P(b)$，在$\mathcal{L}^n$上产生一族概率测度，并且给定一个函数$F \colon \mathcal{L}^n \to \mathbb{R}$，我们的目标是在这些概率测度中找到一个最大化（分别为）的概率测度。最小化）对$F$的期望。在本文中，我们确定了一个家庭这样的功能$F$，所有的期望最大化（分别。在某些条件下最小化）由相同的{ \em积}概率度量定义，该概率度量由称为超级顶点（分别为）的$P(b)$的独特顶点定义。子顶点）。欧式看涨期权的收益属于这类函数。
英文摘要：
The present paper originated from a problem in Financial Mathematics concerned with calculating the value of a European call option based on multiple assets each following the binomial model. The model led to an interesting family of polytopes $P(b)$ associated with the power-set $\mathcal{L} = \wp\{1,\dots,m\}$ and parameterized by $b \in \mathbb{R}^m$, each of which is a collection of probability density function on $\mathcal{L}$. For each non-empty $P(b)$ there results a family of probability measures on $\mathcal{L}^n$ and, given a function $F \colon \mathcal{L}^n \to \mathbb{R}$, our goal is to find among these probability measures one which maximises (resp. minimises) the expectation of $F$. In this paper we identify a family of such functions $F$, all of whose expectations are maximised (resp. minimised under some conditions) by the same {\em product} probability measure defined by a distinguished vertex of $P(b)$ called the supervertex (resp. the subvertex). The pay-offs of European call options belong to this family of functions.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
