Coupling and Control in Continuous Time

随机性在自然世界中无处不在，对随机事件的理解和建模的进步是在自然科学和社会科学、工程、统计学等许多问题上取得进展的关键。耦合是概率论中的一种基本范式，通过这种范式，随机数量（随机变量、随机过程）的概率分布可以通过“逐点”比较相互比较。它产生了分析随机系统的强大技术。马尔可夫过程是一种随机过程，它以现在为条件，它的未来和过去是独立的。也就是说，如果我们知道这个过程的当前状态，我们就不能通过更多地了解它的过去来获得关于它未来演变的额外信息。这种范式描述了自然科学和社会科学中用作模型的许多随机过程。在耦合中，我们看到两个马尔可夫过程，它们从不同的位置开始并共同演化。我们对它们满足一些标准感兴趣，例如，两个过程尽可能快地相遇，尽可能长时间地保持彼此接近，或其他标准（例如，耦合时间的大偏差行为，即耦合时间的指数衰减率是多少）。除了本身是一个有趣的数学问题外，这个问题还具有重要的潜在应用。例如，收敛到随机平衡的速度（在许多应用中是一个关键问题）是由耦合发生的速度控制的。耦合速度有一个自然的下界。“最快”的耦合，即两个过程在任何给定时间内未相遇的概率最小的耦合，被称为“最大”耦合：可以通过将第二个过程定义为第一个过程的整个轨迹的函数来构建这些耦合。然而，在科学建模的背景下，关注共适应耦合是很自然的，即在给定时间的第二个过程只能基于第一个过程的轨迹（即不考虑第一个过程的未来轨迹的信息）来构建的耦合。这里的困难在于很难获得最优（称为“极值”）耦合。事实上，很难知道任何给定的共适应耦合有多好。这个建议是关于采用任何自适应耦合并提供改进它的方法。不仅在局部，而且在数学上证明了我们提出的顺序改进产生了尽可能好的自适应耦合。从本质上讲，我们正在寻找解决一个随机优化问题下的附加约束的共同适应性。在本建议中，改进自适应耦合以达到最优性的主要方法是通过应用控制理论。我们的目标是使用策略改进算法，一个工具从控制理论的工作在离散时间，并发展其在连续时间的应用。在项目的应用部分，我们的目标是开发PIA在非线性偏微分方程理论和具有跳跃过程的多级蒙特卡罗（MLMC）算法中的应用。非线性偏微分方程和MLMC模拟领域具有广泛的社会和经济影响：前者在生物学，物理学，工程等领域有应用，后者在工程和科学的不确定性量化中具有至关重要的意义。当不确定性是高维和强非线性时，蒙特卡罗模拟仍然是首选的方法，在生物化学反应和等离子体物理等多种领域都有应用。

Randomness is ubiquitous in the natural world, and advances in understanding and modelling random events are key to making progress with many problems in the natural and social sciences, engineering, statistics, to name but a few. Coupling is a fundamental paradigm in probability through which probability distributions of random quantities (random variables, random processes) can be compared with each other via "pointwise" comparisons. It yields powerful techniques for analysing random systems. A Markov process is a random process whereby, conditional on the present, its future and past are independent. That is, if we know the present state of the process, we can gain no additional information on its future evolution by knowing more about its past. This paradigm describes many random processes used as models in the natural and social sciences. In coupling we are looking at two Markov processes that start from different locations and evolve jointly. We are interested in them meeting a number of criteria, e.g. the two processes meeting as soon as possible, staying close to each other for as long as possible, or other criteria (e.g. the large deviation behaviour of the coupling time, i.e. what the exponential rate of decay of the coupling time is). As well as being an interesting mathematical question in and of itself, this problem has significant potential applications. For example, the rate of convergence to stochastic equilibrium (a crucial question in many applications) is controlled by the rate at which coupling occurs.There is a natural lower bound in the speed of coupling. The "fastest" couplings, i.e. the couplings where the probability that the two processes have not met by any given time is smallest, are known as "maximal" couplings: one can construct those by defining the second process as a functional of the entire trajectory of the first. However, in the context of modelling in the sciences, it is natural to focus on co-adapted couplings, namely couplings whereby the second process at a given time can only be constructed based on the trajectory of the first upto and including the present time (i.e. no information about the future trajectory of the first process can be taken into account). The difficulty here is that it is hard to obtain optimal (called "extremal") couplings. In fact it's difficult to know how good any given co-adapted coupling is. This proposal is about taking any co-adapted coupling and providing a method of improving it. Not just locally, but proving mathematically that the sequential improvements we propose yield a co-adapted coupling that is as good as it can get. Essentially we are looking to solve a stochastic optimisation problem under the additional constraint of co-adaptivity. In this proposal, the main method for improving a co-adapted coupling to achieve optimality is via the application of control theory. We aim to use the Policy Improvement Algorithm, a tool from control theory that works in discrete time, and develop its application in continuous time. In the application part of the project, we aim to develop applications of the PIA in the theory of non-linear PDEs and Multi-Level Monte Carlo (MLMC) algorithms for processes with jumps. The areas of non-linear PDEs and MLMC simulation have applications with vast societal and economic impact: the former has applications in biology, physics, engineering to name a few, and the latter is of crucial importance in Uncertainty Quantification in engineering and science. When the uncertainty is high-dimensional and strongly nonlinear, Monte Carlo simulation remains the preferred approach, with applications in areas as diverse as biochemical reactions and plasma physics.