论文标题
功能扩展
Functional Expansions
论文作者
论文摘要
在工程，系统理论和金融等许多学科中，路径依赖性无处不在。它反映了过去通常通过功能表达的未来的影响。但是，非马克维亚问题通常是无限维度的，因此从概念和计算的角度来看都有挑战。在这项工作中，我们阐明了功能的扩展。首先，我们处理围绕固定长度的路径进行的静态扩展，并提出Wiener系列$ - $内在价值扩展（IVE）的概括。在动态情况下，我们重新审视功能性泰勒扩展（FTE）。后者将功能性iTOculus与签名连接起来，以量化“扰动”路径与源路径串联时的功能。特别是，FTE优雅地将功能与未来的轨迹分开。实际分析性和收敛半径的概念也扩展到路径空间。我们讨论了希尔伯特（Hilbert）的预测和维纳（Wiener）混乱引起的其他动态扩张，并最终显示了FTE在异国情调索赔的定价和对冲上的财务应用。
Path dependence is omnipresent in many disciplines such as engineering, system theory and finance. It reflects the influence of the past on the future, often expressed through functionals. However, non-Markovian problems are often infinite-dimensional, thus challenging from a conceptual and computational perspective. In this work, we shed light on expansions of functionals. First, we treat static expansions made around paths of fixed length and propose a generalization of the Wiener series$-$the intrinsic value expansion (IVE). In the dynamic case, we revisit the functional Taylor expansion (FTE). The latter connects the functional Itô calculus with the signature to quantify the effect in a functional when a "perturbation" path is concatenated with the source path. In particular, the FTE elegantly separates the functional from future trajectories. The notions of real analyticity and radius of convergence are also extended to the path space. We discuss other dynamic expansions arising from Hilbert projections and the Wiener chaos, and finally show financial applications of the FTE to the pricing and hedging of exotic contingent claims.
