CAREER: Quantification of Risk

风险的量化：摘要风险的概念在保险数学、金融学、可靠性理论、决策论、经济学、最优控制或最优路径规划等领域中发挥着重要的作用。在许多这样的领域，概率和统计方法的应用已经有了很长的历史。为了最大化期望值、最小化方差或控制破产概率，人们花费了大量的精力来开发看似合理的随机模型和技术。对风险的概念、风险的量化及其对监管和行为的影响进行系统调查的工作则少得多。特别是，在动态设置中对风险的一致聚集、风险归属或风险的优化管理的研究才刚刚开始。该项目的目标是为量化和管理不同情况下的风险提供一个数学框架。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类泛函的数学结构，以期获得诸如单调性、凸性或时间一致性等重要性质。此外，还将研究相关优化问题的可处理性、实际应用和数值实现。该项目将涉及概率论、统计学、凸分析、局部凸向量空间和数值数学等方法。该项目探索个人、公司和社会在不同不确定情况下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、缓解和最小化风险提供数学工具。需要处理的实际问题有：最佳投资组合分配；金融衍生品的定价、对冲和设计；股权挂钩保险合同等复杂保险产品的估值；确定银行和保险公司的资本要求；协调一致地确定风险的归属；金融机构之间风险的最佳转移；风险的证券化；综合风险管理；能源合同的估值；发电厂的最佳运作；以及评估社会因金融危机、资金不足的养恤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可望激发新的概念，并在不同领域之间转移专门知识。

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.