In stochastic analysis, a standard method to study a path is to work with its signature. This is a sequence of tensors of different order that encode information of the path in a compact form. When the path varies, such signatures parametrize an algebraic variety in the tensor space. The study of these signature varieties builds a bridge between algebraic geometry and stochastics, and allows a fruitful exchange of techniques, ideas, conjectures and solutions. In this paper we study the signature varieties of two very different classes of paths. The class of rough paths is a natural extension of the class of piecewise smooth paths. It plays a central role in stochastics, and its signature variety is toric. The class of axis-parallel paths has a peculiar combinatoric flavour, and we prove that it is toric in many cases.

中文翻译：

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路径签名品种的复曲面几何

在随机分析中，研究路径的标准方法是使用其签名。这是一个不同阶次的张量序列，以紧凑的形式对路径信息进行编码。当路径变化时，这些签名参数化张量空间中的代数变体。对这些标志性变体的研究在代数几何和随机性之间架起了一座桥梁，并允许对技术、思想、猜想和解决方案进行富有成效的交流。在本文中，我们研究了两类截然不同的路径的签名变体。粗糙路径类是分段平滑路径类的自然延伸。它在随机指标中起着核心作用，其标志性品种是复曲面。轴平行路径类具有特殊的组合风格，我们证明它在许多情况下是复曲面的。