英文标题：
《Density of the set of probability measures with the martingale
representation property》
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作者：
Dmitry Kramkov, Sergio Pulido
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最新提交年份：
2019
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英文摘要：
Let $\\psi$ be a multi-dimensional random variable. We show that the set of probability measures $\\mathbb{Q}$ such that the $\\mathbb{Q}$-martingale $S^{\\mathbb{Q}}_t=\\mathbb{E}^{\\mathbb{Q}}\\left[\\psi\\lvert\\mathcal{F}_{t}\\right]$ has the Martingale Representation Property (MRP) is either empty or dense in $\\mathcal{L}_\\infty$-norm. The proof is based on a related result involving analytic fields of terminal conditions $(\\psi(x))_{x\\in U}$ and probability measures $(\\mathbb{Q}(x))_{x\\in U}$ over an open set $U$. Namely, we show that the set of points $x\\in U$ such that $S_t(x) = \\mathbb{E}^{\\mathbb{Q}(x)}\\left[\\psi(x)\\lvert\\mathcal{F}_{t}\\right]$ does not have the MRP, either coincides with $U$ or has Lebesgue measure zero. Our study is motivated by the problem of endogenous completeness in financial economics.
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中文摘要：
设$\\ psi$为多维随机变量。我们证明了概率测度集$\\mathbb{Q}$，使得$\\mathbb{Q}$-鞅$S^{\\mathbb{Q}}}}t=\\mathbb{E}^{\\mathbb{Q}}左[\\psi\\lvert\\mathcal{F}}{t}}右]$在$\\mathcal L}范数中为空或稠密是的。证明基于一个相关的结果，该结果涉及开放集$U$上的终端条件$（\\ psi（x））{x \\ in U}$和概率测度$（\\ mathbb{Q}（x））{x \\ in U}$的分析域。也就是说，我们证明了U$中的点集$x，使得$S\\t（x）=\\mathbb{E}^{\\mathbb{Q}（x）}\\left[\\psi（x）\\lvert\\mathcal{F}\\ut}\\right]$没有MRP，或者与$U$重合，或者具有Lebesgue测度零。我们的研究是基于金融经济学中的内生完备性问题。
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分类信息：

一级分类：Mathematics 数学
二级分类：Probability 概率
分类描述：Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory
概率论与随机过程的理论与应用：例如中心极限定理，大偏差，随机微分方程，统计力学模型，排队论
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一级分类：Quantitative Finance 数量金融学
二级分类：General Finance 一般财务
分类描述：Development of general quantitative methodologies with applications in finance
通用定量方法的发展及其在金融中的应用
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一级分类：Quantitative Finance 数量金融学
二级分类：Mathematical Finance 数学金融学
分类描述：Mathematical and analytical methods of finance, including stochastic, probabilistic and functional analysis, algebraic, geometric and other methods
金融的数学和分析方法，包括随机、概率和泛函分析、代数、几何和其他方法
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