功能扩展
Functional Expansions
摘要：
路径依赖在工程学、系统论和金融学等学科中无处不在。它反映了过去对未来的影响，通常通过功能来表达。然而，非马尔可夫问题往往是无限维的，因此从概念和计算的角度来看具有挑战性。在这项工作中，我们揭示了扩张的泛函。首先，我们对待静态扩展周围的固定长度的路径，并提出了一个推广的维纳系列$-$的内在值扩展（IVE）。在动态的情况下，我们重新审视功能泰勒展开（FTE）。后者将泛函伊藤微积分与签名联系起来，以量化“扰动”路径与源路径连接时泛函中的影响。特别是，FTE优雅地将泛函与未来轨迹分离。实解析性和收敛半径的概念也被推广到路径空间。我们讨论了其他动态扩展所产生的希尔伯特投影和维纳混沌，最后显示金融应用的FTE奇异未定权益的定价和套期保值。
英文摘要：
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.
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