CAREER: Quantification of Risk

风险量化：风险的概念在保险数学、金融学、可靠性理论、决策理论、经济学、最优控制或最优路径规划等诸多领域中起着重要的作用。在许多这些领域中，概率和统计方法的应用已经有很长的历史。大量的努力已经花费在合理的随机模型和技术的发展，以最大化期望值，最小化方差或控制破产概率。相当少的工作一直致力于一个系统的调查风险的概念，其量化和监管和行为的影响。特别是，在动态设置中，对风险的一致聚合、风险的归因或风险的最佳管理的研究最近才开始。该项目的目标是为不同背景下的风险量化和管理提供一个数学框架。形式上，风险度量是随机变量或随机过程空间上的泛函。该项目的一个重要方面将是研究这种泛函的数学结构，以期实现重要的性质，如单调性，凸性或时间一致性。同时也将研究相关优化问题的易处理性、实际应用和数值实现。这将涉及概率论，统计学，凸分析，局部凸向量空间和数值数学的方法。该项目探讨了个人，公司和社会在不同的不确定性情况下所面临的风险的量化和管理。该项目的主要重点是为评估、控制、缓解和尽量减少风险提供数学工具。需要解决的实际问题是最佳的投资组合分配;金融衍生工具的定价、套期保值和设计;复杂保险产品的估值，如与股票挂钩的保险合同;确定银行和保险公司的资本要求;风险的一致归属;金融机构之间最佳的风险转移;风险证券化;综合风险管理;能源合同的估值;发电厂的最佳运行;以及评估社会因金融危机、资金不足的养老金计划或飓风和地震等自然灾害而面临的风险。对风险的一般处理和对不同具体问题的调查有望激发新的概念和不同领域之间的专业知识转移。

Quantification of Risk: AbstractThe notion of risk plays an important role in various fields such as insurance mathematics, finance, reliability theory, decision theory, economics, optimal control or optimal route planning. In many of these fields, the application of probabilistic and statistical methods has had a long history. A lot of effort has been spent on the development of plausible stochastic models and techniques to maximize expected values, minimize variances or control ruin probabilities. Considerably less work has been devoted to a systematic investigation of the concept of risk, its quantification and the implications for regulation and behavior. In particular, the study of consistent aggregation of risk, attribution of risk, or optimal management of risk in dynamic setups has only started recently. The goal of this project is to provide a mathematical framework for the quantification and management of risk in different contexts. Formally, a risk measure is a functional on a space of random variables or stochastic processes. An important aspect of the project will be the study of the mathematical structure of such functionals with a view towards important properties like monotonicity, convexity or time-consistency. But also the tractability of related optimization problems, practical applications and numerical implementations will be investigated. This will involve methods from probability theory, statistics, convex analysis, locally convex vector spaces and numerical mathematics.The project explores the quantification and management of risks faced by individuals, companies and society in different situations of uncertainty. The main focus of the project is to provide mathematical tools for the assessment, control, mitigation and minimization of risk. Practical problems to be addressed are optimal portfolio allocation; the pricing, hedging and design of financial derivatives; the valuation of complex insurance products such as equity-linked insurance contracts; the determination of capital requirements for banks and insurance companies; coherent attribution of risks; optimal risk transfer between financial institutions; the securitization of risks; integrated risk management; the valuation of energy contracts; the optimal operation of power plants; and the assessment of risk faced by society through financial crises, under-funded pension plans or natural disasters such as hurricanes and earthquakes. The general treatment of risk and the investigation of different concrete problems are expected to motivate new concepts and the transfer of expertise between different fields.