Probabilistic methods in KPZ universality and stochastic optimisation

KPZ 普适性和随机优化中的概率方法

• 批准号: RGPIN-2020-06063
• 负责人: Ortmann, Janosch
• 金额: $ 1.68万
• 依托单位: Université du Québec à Montréal
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2020
• 资助国家: 加拿大
• 起止时间: 2020-01-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=718226
• 关键词: Probabilistic; methods; KPZ; universality; stochastic

The proposed research is situated in the field of probability theory, that is the study of phenomena under uncertainty. The global objective is to better understand random occurrences, particularly their distribution and their long-term behaviour. In this proposal, we are concerned both with theoretical and applied aspects of this field. A common theme is the quest to optimise a quantity subject to random constraints, and the need to study the influence and benefits of uncertainty in this setting. To give a specific example of the kind of phenomena studied, consider last-passage percolation, which can be considered as a model for the movement of liquids through a porous environment: a walker crosses a grid, with random numbers of rewards placed on each point of the grid. The aim of the walker is to choose a path so as to maximise the total reward she picks up along the way. If the rewards are placed randomly, subject to a given probability distribution, the optimal path and the largest possible rewards are themselves random variables. Studying these random variables, particularly as the distance between start and end-point becomes larger, is a key objective of this proposal. A surprising finding is that this long-term behaviour doesn't depend on the specific reward distribution chosen. This is an example of a phenomenon called universality and has been observed in liquid crystal growth, bacterial colony growth and fire propagation. A surprising array of mathematical techniques have been applied to its study, including combinatorics, algebra, analysis and queueing theory. A related problem concerns finding the minimum of a quantity, for example the cost of transporting goods across a network, subject to random constraints, such as the demand for goods to be transported. This problem is often solved numerically (with the help of computer programs) and the uncertain parameters are represented by a finite number of scenarios. However, in order to accurately represent the underlying uncertainty, a large number of scenarios must be sampled. Apart from being computationally expensive, this high complexity can also make it difficult for decision makers to understand how a particular decision was chosen by the model. In this proposal, I will produce new trade-offs between complexity and accuracy. A crucial ingredient are opportunity cost, that is the cost of making a decision after predicting the wrong scenario. Similarly, managers of large infrastructure projects often need to make optimal investment decisions based on uncertain future cash flows. In order to help them make better decisions, it has proved to be helpful to model the possibility to invest as a financial option. While there are similarities to the theory of financial options, there are significant differences, because the two follow very different assumptions. In the proposed research, I will contribute to a better understanding of this model with innovative probabilistic approaches.

该研究属于概率论领域，即研究不确定性下的现象。全球目标是更好地了解随机事件，特别是其分布和长期行为。在这一建议中，我们关注的是这一领域的理论和应用方面。一个共同的主题是寻求优化的数量受到随机约束，并需要研究在这种情况下的不确定性的影响和好处。 为了给出所研究的现象的一个具体例子，考虑最后一次通过渗透，它可以被认为是液体通过多孔环境的运动模型：一个步行者穿过网格，网格的每个点上放置随机数量的奖励。步行者的目标是选择一条路径，以便最大限度地提高她沿着获得的总回报。如果奖励是随机放置的，服从给定的概率分布，最佳路径和最大可能的奖励本身就是随机变量。研究这些随机变量，特别是当起点和终点之间的距离变大时，是本提案的一个关键目标。一个令人惊讶的发现是，这种长期行为并不取决于所选择的特定奖励分配。这是一个被称为普遍性的现象的例子，已经在液晶生长、细菌菌落生长和火灾传播中观察到。一系列令人惊讶的数学技术已被应用于其研究，包括组合学，代数，分析和推理理论。 一个相关的问题涉及找到数量的最小值，例如在网络上运输货物的成本，受到随机约束，例如对运输货物的需求。这个问题往往是解决数值（与计算机程序的帮助下）和不确定的参数表示的有限数量的情况。然而，为了准确地表示潜在的不确定性，必须对大量情景进行抽样。除了计算成本高之外，这种高复杂性还使决策者难以理解模型如何选择特定的决策。在这个建议中，我将在复杂性和准确性之间做出新的权衡。一个关键因素是机会成本，即在预测错误的情况后做出决策的成本。 同样，大型基础设施项目的管理者往往需要根据不确定的未来现金流做出最佳投资决策。为了帮助他们做出更好的决策，事实证明，将投资的可能性作为一种金融选择进行建模是有帮助的。虽然与金融期权理论有相似之处，但也有重大差异，因为两者遵循非常不同的假设。在拟议的研究中，我将有助于更好地理解这个模型与创新的概率方法。