Analysis and control of path-dependent random systems

随机过程的路径依赖泛函的研究是概率论中的一个重要课题，在数学物理、最优控制理论、种群动力学和数学金融中都有广泛的应用。另一方面，可用于模拟和分析随机系统的绝大多数理论工具侧重于“马尔可夫系统”--没有过去记忆的随机系统--它们更容易分析和模拟，而且有广泛的分析结果和工具可用。这个项目的目标是利用随机分析的最新发展--特别是由Rama Cont和合作者近年来开发的‘Function Ito Calculus’--为具有路径依赖特征的随机系统的灵敏度分析和最优控制开发一个数学框架。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，该框架由Rama Cont和合作者在Föllmer早期工作的基础上开发，它将经典的Ito演算扩展到一大类依赖于路径的泛函。这种路径演算和它的概率对应的泛函伊藤演算目前是随机分析中一个新兴文献的焦点，它开始被探索，并指出与先前建立的数学概念，如粗糙路径理论，向前-向后随机微分方程和Malliavin演算，取得了丰硕的联系。泛函Ito演算将马尔可夫过程与抛物型偏微算子之间的关系推广到具有路径依赖特征的非马尔可夫过程的情形，并在连续函数空间上得到了一类新的泛函偏微分方程组，称为路径依赖的Kolmogorov方程。这项建议的目的是对泛函Ito演算和路径依赖的Kolmogorov方程进行系统的研究，它们在分析和概率论中的分支，以及它们在具有路径依赖特征的系统的灵敏度分析和最优控制中的应用。我们将探讨非预期路径演算及其与粗糙路理论的关系、泛函Ito演算及其与Malliavin演算的关系、不连续过程的泛函、依赖于路径的Kolmogorov方程的解的各种概念、在非马尔科夫随机控制和倒向随机微分方程中的应用以及在鞅表示中的应用。拟议研究的一个主要目标是将随机过程和偏微分方程之间的联系扩展到经典的马尔科夫环境之外，使以前仅限于对马尔可夫现象进行建模的一系列分析工具适用于在物理、工程和生命科学的各种应用中可能出现的广泛的(非马尔科夫)随机过程。通过将这些分析工具的适用性扩展到更广泛的问题，我们的研究旨在使更多的科学家、工程师和最终用户可以使用它们来模拟、灵敏度分析和对具有路径依赖特征的系统进行最优控制。这些问题在随机路径依赖系统的分析、仿真和最优控制中有着广泛的应用，如物理学、数理金融学和种群动力学等。我们尤其打算详细地探索该理论的计算方面，并开发用于模拟和分析具有路径依赖特征的随机系统的开源软件工具。致力于有记忆系统的最优控制、种群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，并从中受益。

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.