Analysis and control of path-dependent random systems

随机过程的路径依赖泛函（英语：Path-dependent functional of random processes）是概率论中的一个重要课题，在数学物理学、最优控制理论、人口动力学和数学金融学中有着广泛的应用。另一方面，可用于随机系统模拟和分析的绝大多数理论工具都集中在“马尔可夫系统”上--没有过去记忆的随机系统--这更容易分析和模拟，并且有各种各样的分析结果和工具可供使用。该项目的目标是利用随机分析的最新发展-特别是拉马康特和合作者近年来开发的“功能伊藤演算”-开发具有路径依赖特征的随机系统的灵敏度分析和最优控制的数学框架。 这种方法的一个关键组成部分是一个被称为“非预期泛函演算”的数学框架，由拉马·康特和合作者基于Föllmer的开创性早期工作开发，将经典的伊藤演算扩展到一大类路径依赖泛函。这pathwise演算，其概率对应，功能伊藤演算，目前的重点是一个新兴的文学在随机分析，这是开始进行探索，并指出富有成效的联系与以前建立的数学概念，如粗糙路径理论，向前向后随机微分方程和Malliavin演算。泛函伊藤演算将马尔可夫过程和抛物型偏微分算子之间的关系扩展到具有路径依赖特征的非马尔可夫过程的情况，并导致一类新的泛函偏微分方程在连续函数空间上，称为“路径依赖Kolmogorov方程”。本建议的目标是进行系统的研究功能伊藤演算和路径依赖的Kolmogorov方程，其分支在分析和概率论及其应用的灵敏度分析和最优控制系统的路径依赖功能。我们将探索非预期路径演算及其与粗糙路径理论的关系，功能伊藤演算及其与Malliavin演算的关系，不连续过程的泛函，路径相关Kolmogorov方程的各种解的概念，非马尔可夫随机控制和倒向随机微分方程的应用，以及鞅表示的应用。拟议的研究的一个主要目标是扩展随机过程和偏微分方程之间的联系，超越经典的马尔可夫设置，使一系列的分析工具，以前仅限于适用于广泛的（非马尔可夫）随机过程的马尔可夫现象的建模可能会出现在各种类型的应用在物理学，工程和生命科学。通过将这些分析工具的适用性扩展到更广泛的问题，我们的研究旨在使它们可用于更广泛的科学家，工程师和最终用户，用于模拟，灵敏度分析和具有路径依赖特征的系统的最优控制。这些问题在物理学、数学金融学和人口动力学中的随机路径依赖系统的分析、模拟和最优控制中有着广泛的应用。我们打算特别是详细探讨理论的计算方面，并开发开源软件工具，用于模拟和分析具有路径依赖特征的随机系统。研究记忆系统最优控制、人口动态模型和数学金融的科学家和工程师可能会对这项研究感兴趣/从中受益。

The study of path-dependent functionals of random processes -quantities which depend on the path of a random process- is an important topic in probability and arises in various applications of probability, in mathematical physics, optimal control theory, population dynamics and mathematical finance. On the other hand, the vast majority of theoretical tools available for the simulation and analysis of stochastic systems focus on "Markovian systems" -random systems with no memory of the past- which are easier to analyse and simulate and for which a wide array of analytical results and tools are available. The ambition of this project is to capitalize on recent developments in stochastic analysis -in particular the 'Functional Ito Calculus' developed by Rama Cont and collaborators in the recent years- to develop a mathematical framework for sensitivity analysis and optimal control of random systems with path-dependent features. A key ingredient of this approach is a mathematical framework known as the "non-anticipative functional calculus", developed by Rama Cont and collaborators based on pioneering early work of Föllmer, which extends the classical Ito calculus to a large class of path-dependent functionals. This pathwise calculus, and its probabilistic counterpart, the Functional Ito Calculus, are currently the focus of a burgeoning literature in stochastic analysis, which is starting to be explored and which points to fruitful links with previously established mathematical concepts such as Rough Path theory, Forward-Backward Stochastic Differential Equations and Malliavin Calculus. The Functional Ito calculus extends well-known relations between Markov processes and parabolic partial differential operators to the case of non-Markovian processes with path-dependent features, and leads to a new class of functional partial differential equations on the space of continuous functions, called 'path-dependent Kolmogorov equations'. The goal of this proposal is to undertake a systematic study of the Functional Ito Calculus and path-dependent Kolmogorov equations, their ramifications in analysis and probability theory and their applications to the sensitivity analysis and optimal control of systems with path-dependent features. We will explore the non-anticipative pathwise calculus and its relation with rough path theory, functional Ito calculus and its relation with the Malliavin calculus, functionals of discontinuous processes, various notions of solutions for path-dependent Kolmogorov equations, applications to non-Markovian stochastic control and Backward Stochastic Differential Equations, and application to Martingale Representation. A key ambition of the proposed research is to extend the links between stochastic processes and partial differential equations beyond the classical Markovian setting, making a range of analytical tools, formerly restricted to the modelling of Markovian phenomena applicable to a wide range of (non-Markovian) stochastic processes which may arise in various types of application in physics, engineering and the life sciences. By expanding the applicability of these analytical tools to a wider array of problems, our research aims to make them available to a wider array of scientists, engineers and end-users for the purpose of simulation, sensitivity analysis, and optimal control of systems with path-dependent features. These questions have numerous applications to the analysis, simulation and optimal control of random path-dependent systems in physics, mathematical finance and population dynamics. We intend in particular to explore in detail the computational aspects of the theory and develop open-source software tools for the simulation and analysis of stochastic systems with path-dependent features. Scientists and engineers working on optimal control of systems with memory, models of population dynamics, and mathematical finance are likely to be interested in/to be benefited from the research.