Wasserstein与适应Wasserstein分布鲁棒优化问题的高阶灵敏度分析

中文摘要 AI 辅助

我们研究了概率测度泛函灵敏度的高阶展开。基于bartl2021sensitivity和bartlsensitivityadapted的工作（他们建立了当泛函源自随机优化问题时的一阶展开），我们利用Wasserstein空间微分学的最新进展扩展了这些结果。具体而言，我们将他们的方法推广到定义在Wasserstein空间上的更广泛泛函类别。我们提供了关于Wasserstein和适应Wasserstein度量下泛函灵敏度的二阶展开的显式公式。此外，我们提出了这些灵敏度高阶展开的构造方法。最后，借鉴Touzisauldubois2024ordermartingalemodelrisk和JiangObloj关于鞅约束下分布鲁棒优化灵敏度的最新进展，我们还推导了在存在此类约束时灵敏度的高阶展开。

英文摘要

We investigate the higher-order expansions of the sensitivities of functionals of probability measures. Building on the work of \citeauthor{bartl2021sensitivity} \cite{bartl2021sensitivity} and \citeauthor{bartlsensitivityadapted} \cite{bartlsensitivityadapted}, who established the first-order expansion when the functional arises from a stochastic optimisation problem, we extend these results using recent developments in differential calculus on Wasserstein spaces. Specifically, we generalise their approach to broader classes of functionals defined on the Wasserstein space. We provide an explicit formula for the second-order expansion of the sensitivity of a functional with respect to both the Wasserstein and the adapted Wasserstein metrics. Furthermore, we propose a construction method for the higher-order expansion of these sensitivities. Finally, following recent advances by \citeauthor{Touzisauldubois2024ordermartingalemodelrisk} \cite{Touzisauldubois2024ordermartingalemodelrisk} and \citeauthor{JiangObloj} \cite{JiangObloj} concerning sensitivity of Distributionally Robust Optimisation under martingale constraints, we also derive higher-order expansions for sensitivities in the presence of such constraints.

发表机构

• New York University, Tandon School of Engineering（纽约大学坦登工程学院）

机构由 AI 辅助整理，请以论文原文为准。