Novel stochastic models in risk management and game theory

***随机模型对于解决定量金融和经济学中的各种问题非常有用。然而，金融和经济学的最新发展带来了新的问题，例如，金融市场的合适基准是什么，或者私人信息如何影响经济决策。本研究的目的是通过引入新的随机模型和开发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。拟定的研究计划由以下四部分组成：******1。部分：基准的交易和均衡模型***最近有关操纵金融市场价格基准的丑闻引发了有关此类基准设计的激烈政策讨论。在提出的研究中，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。部分：具有私人监控的连续时间博弈***企业通常既拥有来自市场的信息，也拥有自己的私人信息，而不直接观察其他企业的行为。这种情况导致了所谓的带有私人监控的游戏。我们计划在连续时间设置中分析这样的博弈，用不同的相关信号来模拟公司的不同信息。******3。第一部分：银行网络形成的博弈论模型***自2007/8年金融危机以来，银行如何通过贷款和金融衍生品相互联系的研究受到了广泛关注。在提出的研究计划的这一部分中，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，从而形成网络上重复博弈的公式。******4。部分：信用风险管理中的机器学习***来自机器学习的技术越来越多地被银行用于估计其个人客户的信用风险。在最后一部分中，我们开发了一个数学框架，可以应用于从个人客户的数据分析到汇总风险措施的统计推断，这对于确定银行的资本要求很重要。******拟议的研究在三个方面使加拿大受益：通过开发新颖的随机模型使加拿大的数学和统计研究界受益；通过为解决专题问题提供健全的数学框架使加拿大的金融行业受益；通过为与学术界和工业界高度相关的问题的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.