Novel stochastic models in risk management and game theory

人们发现，随机模型在解决定量金融和经济学中的各种问题时非常有用。然而，最近金融和经济学的发展带来了新的问题，例如，金融市场中什么是合适的基准，或者私人信息如何影响经济决策。这项研究的目的是通过引入新的随机模型和发展随机和控制理论中的相关数学工具来解决风险管理和博弈论中的这些问题。拟议的研究计划包括以下四个部分。1.基准的交易和均衡模型最近金融市场操纵价格基准的丑闻引发了关于设计此类基准的激烈政策讨论。在拟议的研究中，我们开发了一个数学框架，该框架捕获了基准适用性的主要标准。2.第二部分：私营监测公司的持续博弈往往既有来自市场的信息，也有自己的私人信息，而不是直接观察其他公司的行为。这种情况导致了所谓的私人监控游戏。我们计划在具有不同相关信号的连续时间环境中分析这样的博弈，以模拟公司的不同信息。3.第三部分：银行网络形成的博弈论模型自2007/8金融危机以来，关于银行如何通过贷款和金融衍生品相互联系的研究受到了极大的关注。在拟议的研究计划的这一部分，我们开发了一个动态模型来研究银行如何随着时间的推移调整彼此之间的风险敞口，从而导致网络上的重复博弈。第四部分：机器学习在信用风险管理中的应用来自机器学习的技术越来越多地被银行用于估计个人客户的信用风险。在最后这一部分，我们开发了一个数学框架，可以用于从对个别客户的数据分析到综合风险衡量标准进行统计推断，这在确定银行的资本要求方面非常重要。拟议的研究在三个方面使加拿大受益：通过开发新的随机模型使加拿大的数学和统计研究界受益，通过提供解决专题问题的完善的数学框架使其金融业受益，通过为与学术界和工业界高度相关的问题进行数学和统计建模提供极好的培训机会使加拿大的学生受益。

Stochastic models have been found extremely useful to address various problems in quantitative finance and economics. However, recent developments in finance and economics led to new questions, such as, what suitable benchmarks in financial markets are, or how private information affects economic decisions. The aim of the proposed research is to address such questions in risk management and game theory by introducing new stochastic models and developing related mathematical tools in stochastics and control theory. The proposed research program consists of the following four parts. 1. Part: Trading and Equilibrium Models for Benchmarks Recent scandals over manipulation of price benchmarks on financial markets led to intense policy discussions on the design of such benchmarks. In the proposed research, we develop a mathematical framework that captures main criteria for suitability of benchmarks. 2. Part: Continuous-Time Games with Private Monitoring Firms often have both information from the market and their own private information while not directly observing other firms' actions. Such situations lead to so-called games with private monitoring. We plan to analyze such games in a continuous-time setting with different, correlated signals to model the different information of the firms. 3. Part: A Game-Theoretic Model for Banking Network Formation Since the financial crisis in 2007/8, the study of how banks are linked to each other via loans and financial derivatives has gained a lot of attention. In this part of the proposed research program, we develop a dynamic model to study how banks adjust their exposures between each other over time, leading to a formulation of repeated games on networks. 4. Part: Machine Learning in Credit Risk Management Techniques from machine learning are increasingly often applied by banks to estimate the credit risk of their individual customers. In this last part, we develop a mathematical framework that can be applied to make statistical inferences from data analytics on individual customers to aggregate risk measures, which are important in determining banks' capital requirements. The proposed research benefits Canada in three ways: its mathematical and statistical research community by developing novel stochastic models, its financial industry by providing a sound mathematical framework to address topical questions, and its students by giving excellent training opportunities in mathematical and statistical modelling of problems that are highly relevant in academia and industry.