Novel stochastic models in risk management and game theory

随机模型被发现对解决数量金融学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，金融市场的适当基准是什么，或者私人信息如何影响经济决策。建议的研究的目的是解决这些问题，在风险管理和博弈论引入新的随机模型和发展相关的数学工具，随机和控制理论。本文的研究方案主要包括以下四个部分。 1.部分：基准的交易和均衡模型 最近关于操纵金融市场价格基准的丑闻导致了关于设计这种基准的激烈的政策讨论。在拟议的研究中，我们开发了一个数学框架，捕捉基准的适用性的主要标准。 2. Part：有私人监控的连续时间博弈 企业往往既有来自市场的信息，又有自己的私人信息，而不直接观察其他企业的行动。这种情况导致了所谓的私人监控游戏。我们计划在一个连续的时间设置中分析这种游戏，不同的，相关的信号来模拟不同的公司信息。 3.第一部分：银行网络形成的博弈模型 自2007/08年金融危机以来，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广泛关注。在这一部分的研究计划中，我们开发了一个动态模型来研究银行如何调整他们的风险暴露在彼此之间随着时间的推移，导致制定重复的游戏网络。 4.部分：信用风险管理中的机器学习 银行越来越多地使用机器学习技术来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度量进行统计推断，这对于确定银行的资本要求非常重要。 拟议的研究有利于加拿大在三个方面：其数学和统计研究社区通过开发新的随机模型，其金融业提供了一个健全的数学框架，以解决专题问题，并通过提供良好的培训机会，在数学和统计建模的问题，是在学术界和工业界高度相关的学生。

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.