Stochastic Control under Model Uncertainty

模型不确定性问题最近在应用数学中受到了广泛的关注，特别是在被确定为最近信贷危机的一个原因之后。研究者致力于开发不确定性下随机控制问题的概率工具，特别是数学金融和相关的博弈论问题。重点是在模型不确定性严重的情况下，无法构建参考概率度量，因此不能应用随机分析的标准工具。本项目的第一部分研究了模型不确定性下欧洲可能路径依赖期权的超套期保值问题，以及交易策略和（可能单一的）风险中性措施之间的基本二元性。第二部分研究控制与停止的随机微分对策，并在一个特殊情况下研究不确定条件下美式期权的定价问题。第三部分研究了Doob-Meyer分解的非线性版本，这是在不确定条件下半鞅和随机积分理论的一步。该项目寻求对模型不确定性的更好的数学理解。其结果有助于解决数学金融中定价、风险度量和对冲的重要问题。此外，该项目导致随机分析和控制理论的方法学发展。

The problem of model uncertainty has recently received widespread attention in applied mathematics, especially after being identified as a cause for the recent credit crisis. The investigator seeks to develop probabilistic tools for stochastic control problems under uncertainty, in particular in mathematical finance and related game theoretic problems. The focus is on situations of severe model uncertainty, where no reference probability measure can be constructed and, therefore, standard tools from stochastic analysis cannot be applied. The first part of this project studies the problem of superhedging for a European, possibly path-dependent option under model uncertainty as well as the fundamental duality between trading strategies and (possibly singular) risk-neutral measures. The second part is concerned with stochastic differential games of control and stopping and, in a special case, the pricing of American options under uncertainty. The third part investigates a nonlinear version of the Doob-Meyer decomposition, which is a step towards a theory of semimartingales and stochastic integration under uncertainty. This project seeks a better mathematical understanding of model uncertainty. Its results help to resolve important questions of pricing, risk measurement, and hedging in mathematical finance. Moreover, the project leads to methodological developments in the theory of stochastic analysis and control.