The Functional Ito Calculus and Applications to Causal Optimal Transport

函数伊藤演算及其在因果最优传输中的应用

• 批准号: 2580705
• 金额: --
• 依托单位: University of Oxford
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2021
• 资助国家: 英国
• 起止时间: 2021 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2580705
• 关键词: Functional; Ito; Calculus; Applications; Causal

This research project seeks to study the methods arising from the Functional Ito's Calculus, developed by Cont et al.[1-3], and consider its application to Causal Optimal Transport problems introduced in [4]. The theory of optimal transport began as an engineering problem considered by Monge (and later by Kantorovich), on how to most efficiently transport mass from one location to another. Over the last few decades, the theory that spawned from this deceptively simple problem has experienced fervent development after connections to several different areas of mathematics were made[5]. The field of stochastic analysis, which is the study of processes that evolve randomly with respect to time, has been no exception to this[6,7]. A variant of the Optimal Transport problem, the Causal Optimal Transport problem and its associated notions are born out of a desire to capture the temporal structure of stochastic processes in the context of Optimal Transport and has already found use in answering questions from stochastic control[4,8] and mathematical finance[9]. However, Causal Optimal Transport is a relatively new concept and while some work has been done for the problem in both discrete time[10] and continuous time[8], there are still many open questions and possible points of inquiry. For example, one might ask which results in classical Optimal Transport theory have a "causal" counterpart. We intend to tackle these unanswered questions with the help of Functional Ito's Calculus - a non-anticipative functional calculus which generalizes several well-known results from classical Ito's Calculus[11] to path-dependent functionals of stochastic processes[1,12]. This presents a novel and promising avenue of research. Similar to the Causal Optimal Transport problem, Functional Ito's Calculus has also found use in the fields of stochastic control[13] and mathematical finance[14]. Indeed, Functional Ito's Calculus has, among other examples, already led to the formulation of a new class of partial differential equations - functional Kolmogorov equations[12], a generalization of the well-known Kolmogorov equations which characterizes Markov processes [11,15]. It is then hoped that the efficacy of Functional Ito's Calculus in advancing other areas of research will persist in the context of the Causal Optimal Transport problem and lead to fruitful results.This project falls within the EPSRC Mathematical Analysis research area.

本研究试图研究由Cont等人[1-3]发展的泛函Ito演算产生的方法，并考虑它在[4]中介绍的因果最优运输问题中的应用。最优运输理论最初是蒙格(后来康托洛维奇)考虑的一个工程问题，关于如何最有效地将质量从一个地点运送到另一个地点。在过去的几十年里，从这个看似简单的问题中产生的理论在与几个不同的数学领域建立联系后经历了迅猛的发展[5]。随机分析领域也不例外，它研究随时间随机演变的过程[6，7]。作为最优运输问题的变体，因果最优运输问题及其相关概念产生于在最优运输的背景下捕捉随机过程的时间结构的愿望，并且已经被用于回答随机控制[4，8]和数学金融[9]中的问题。然而，因果最优运输是一个相对较新的概念，虽然已经对离散时间[10]和连续时间[8]的问题做了一些工作，但仍然有许多开放的问题和可能的探索点。例如，有人可能会问，在经典的最优运输理论中，哪些结果具有“因果”对应关系。我们打算借助泛函Ito演算来解决这些悬而未决的问题，这是一种非预期的泛函演算，它将经典的Ito演算[11]中的几个著名结果推广到随机过程的路径相关泛函[1，12]。这为研究提供了一条新颖而有前景的途径。与因果最优运输问题类似，泛函伊藤演算也被用于随机控制[13]和数学金融[14]领域。事实上，泛函伊藤微积分在其他例子中已经导致了一类新的偏微分方程的形成-泛函Kolmogorov方程[12]，它是著名的表征马尔可夫过程的Kolmogorov方程的推广[11，15]。因此，人们希望函数伊藤演算在推进其他研究领域的有效性将继续存在于因果最优运输问题的背景下，并导致丰硕的结果。本项目属于EPSRC数学分析研究领域。