Dynamic Monetary Risk Measures

在单周期风险度量中，风险对象是由实值随机变量的线性空间的元素建模的未来净值，而风险度量是从这个随机变量空间到实数的映射。未来净资产的典型例子是企业权益的会计价值、保险公司的盈余或金融证券投资组合的市场价值。在动态设置中，人们可以模拟具有随机过程的网络的未来演变，并且可以在初始和以后的时间计算风险。如果将相同的风险分配给未来的财务状况，而不管该风险是直接计算出来的，还是按时间回溯的两步计算出来的，则这种风险度量过程被称为时间一致性。本项目的目的是了解货币风险措施的时间一致过程的结构，找到好的例子并研究它们在计算机上的实施。它涉及随机过程的一般理论方法，局部凸向量空间以及线性和凸优化技术。货币风险措施在形式上与不完全市场中的定价、约束下的套期保值、最大期望效用函数、合作博弈论中的精确博弈以及不精确概率理论中的连贯较低预测密切相关。由于在这些理论中，动力学方面还没有详细阐述，对货币风险措施的动力学研究也将在数学和经济学的其他领域产生新的结果。所提出的工作有望导致一类新的风险度量和算法的数值计算。新的风险措施将具有时间一致性，并且在定价、风险资本分配和公平溢价计算方面具有良好的性能。这将为银行业和保险业的监管、企业的内部风险管理以及通过养老金福利担保公司（Pension Benefit Guaranty Corporation）等联邦机构对社会承担的风险进行现实评估带来更好的方法。

In one-period risk measurement, the risky objects are futurenet worths modelled by elements of a linear space of real-valued random variablesand a risk measure is a mapping from this space of random variables to the real numbers.Typical examples of future net worths are the accounting value of afirm's equity, the surplus of an insurance company or the marketvalue of a portfolio of financial securities.In a dynamic setup one can model the future evolution of networths with stochastic processes and risks can be calculated atinitial and later times. A process of risk measures is called time-consistent ifit assigns to a future financial position the same risk irrespective of whether it iscalculated directly or in two steps backward in time. The purpose of this projectis to understand the structure of time-consistent processes of monetary risk measures,find good examples and study their implementation on a computer. It involves methodsfrom the general theory of stochastic processes, locally convex vector spaces aswell as techniques from linear and convex optimization.Monetary risk measures are formally strongly relatedto pricing in incomplete markets, hedging under constraints,maxmin expected utility functionals,exact games in cooperative game theory as well as tocoherent lower previsions in the theory of imprecise probabilities.Since in most of these theories the dynamical aspects are not elaboratedyet, research on the dynamics of monetary risk measures willalso produce new results in other areas of mathematics and economics.The proposed work is expected to lead to a new class of riskmeasures and algorithms for their numerical calculations. The newrisk measures will be time-consistent and have good properties for the purposesof pricing, allocation of risk capital and fair premium calculation. This will leadto better methods for the regulation of the banking and insuranceindustry, the internal risk management of corporations and arealistic assessment of risks the society bears through federalagencies like the Pension Benefit Guaranty Corporation, whichguarantees pension benefits promised by insolvent pension plans.