The Functional Ito Calculus and Applications to Causal Optimal Transport

本研究项目旨在研究由Cont等人开发的伊藤函数微积分所产生的方法。[1-3]，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由蒙日（后来由康托洛维奇）考虑的一个工程问题，关于如何最有效地将质量从一个位置运输到另一个位置。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激烈的发展[5]。随机分析领域，即研究随时间随机演化的过程，也不例外[6，7]。最优运输问题的一个变体，因果最优运输问题及其相关概念诞生于在最优运输的背景下捕获随机过程的时间结构的愿望，并且已经在回答随机控制[4，8]和数学金融[9]的问题中找到了用途。然而，因果最优运输是一个相对较新的概念，虽然在离散时间[10]和连续时间[8]中对该问题做了一些工作，但仍然有许多开放的问题和可能的调查点。例如，人们可能会问，经典最优运输理论中的哪些结果具有“因果”对应。我们打算在函数伊藤微积分的帮助下解决这些未回答的问题-一种非预期的函数微积分，它将经典伊藤微积分[11]的几个著名结果推广到随机过程的路径依赖泛函[1，12]。这提出了一个新的和有前途的研究途径。与因果最优运输问题类似，泛函伊藤微积分也在随机控制[13]和数学金融[14]领域中得到了应用。事实上，伊藤泛函微积分已经导致了一类新的偏微分方程-泛函柯尔莫哥洛夫方程[12]的形成，这是众所周知的柯尔莫哥洛夫方程的推广，它表征了马尔可夫过程[11，15]。因此，希望伊藤函数演算在推进其他领域的研究中发挥作用，并在因果最优运输问题的研究中取得丰硕成果。本项目属于EPSRC数学分析研究领域的福尔斯。

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.