论文标题
通用近似定理，用于Càdlàg路径和莱维型签名模型的连续函数
Universal approximation theorems for continuous functions of càdlàg paths and Lévy-type signature models
论文作者
论文摘要
我们证明了一个通用近似定理，该定理允许通过其时间扩展的签名的线性函数在时间和紧凑的路径集上均匀地近似Càdlàg（粗糙）路径的连续函数。我们处理这个问题的主要动机来自基于签名的金融模型，允许加入跳跃。确实，作为一个重要的应用程序，我们根据增强的莱维过程定义了新的通用签名模型，我们称之为莱维型签名模型。他们扩展了根据建议的资产价格的连续签名模型，例如Arribas等人（2020）在几个方向上，同时仍然保留了普遍性和障碍性能。为了分析这一点，我们首先表明，通用多元Lévy过程的签名过程是扩展张量代数的多项式过程，然后将其用于Lévy-type签名模型中的定价和对冲方法。
We prove a universal approximation theorem that allows to approximate continuous functionals of càdlàg (rough) paths uniformly in time and on compact sets of paths via linear functionals of their time-extended signature. Our main motivation to treat this question comes from signature-based models for finance that allow for the inclusion of jumps. Indeed, as an important application, we define a new class of universal signature models based on an augmented Lévy process, which we call Lévy-type signature models. They extend continuous signature models for asset prices as proposed e.g. by Arribas et al.(2020) in several directions, while still preserving universality and tractability properties. To analyze this, we first show that the signature process of a generic multivariate Lévy process is a polynomial process on the extended tensor algebra and then use this for pricing and hedging approaches within Lévy-type signature models.
