Some Mathematical Finance Problems Under Model Uncertainty

模型不确定性下的金融问题的研究近年来重新引起人们的关注。从实践的角度来看，没有一个单一的预定模型可以完全描述一个复杂的金融市场;考虑到所有与市场数据兼容的概率模型更合理。考虑模型不确定性的另一个原因在于，不同的投资者可能对市场如何演变有不同的信念或预测，对相同的刺激做出不同的反应。在本研究计画中，研究者将探讨模型不确定性下的数个随机最佳化问题，这些问题与博奕论中的微分与纳什均衡、分位数避险以及有交易成本的衍生产品定价有关。这些研究旨在阐明模型不确定性的某些方面，这些方面不仅可以应用于金融市场，也可以应用于其他动力系统。在数学上，模型的不确定性通常由一组非支配的相互奇异的概率表示（在一个概率下可以忽略的集合在另一个概率下可以具有正质量）。在非支配概率集上，研究了不确定性评价准则变化时的四个随机优化问题：（1）确定鲁棒Dynkin对策是否有值并存在最优三元组，（2）确定多个局中人之间的鲁棒非零和对策是否存在Nash均衡，（3）确定鲁棒分位数对冲问题的最优策略，（4）确定鲁棒非零和对策是否存在最优三元组，（5）确定鲁棒非零和对策是否存在最优三元组，（6）确定鲁棒非零和对策是否存在Nash均衡，（7）确定鲁棒分位数对冲问题的最优策略，（8）确定鲁棒非零和对策是否存在最优三元组，（9）确定鲁棒非零和对策是否存在最优三元组，（10）确定鲁棒非零和对策是否存在最优三元组，（10）确定鲁棒非零和对策是否存在最优三元组，（10）确定鲁棒非零和对策是否存在最优三元组，（11）确定鲁棒非零和对策是否存在最优（4）对有交易费用的资产定价基本定理的鲁棒连续时间形式，给出了无套利条件.由于非支配概率集中缺少参考概率，使得经典的工具，如支配收敛定理和Komlos分离引理，很难用来分析与概率集相关的非线性期望。该项目旨在开发新的方法来处理模型不确定性下更复杂的概率情况。这些都是有用的，在更一般的主题，随机控制和优化。

The study of financial questions under model uncertainty has recently attracted renewed attention. From a practical point of view, no single predetermined model can fully describe a complicated financial market; it is more reasonable to take into account all probabilistic models that are compatible with market data. Another reason to consider model uncertainty lies in the fact that different investors may have different beliefs or predictions about how the market would evolve, reacting differently to the same stimuli. In this research project, the investigator studies several stochastic optimization problems under model uncertainty that are related to differential and Nash equilibrium in game theory, quantile hedging, and derivative pricing with transaction costs. These studies aim to illuminate some aspects of model uncertainty, which can be applied not only to financial markets but also to other dynamical systems. Mathematically, model uncertainty is usually represented by a non-dominated set of mutually singular probabilities (a negligible set under one probability can have positive mass under another probability). The investigator analyzes four stochastic optimization problems when the uncertain evaluation criterion varies in the nondominated probability set: (1) to determine whether a robust Dynkin game has a value and admits an optimal triplet; (2) to determine whether a robust non-zero-sum game among many players has a Nash equilibrium; (3) to find an optimal strategy for a robust quantile-hedging problem; and (4) to find a no-arbitrage condition for a robust continuous-time form of the fundamental theorem of asset pricing with transaction costs. The lack of a reference probability in the nondominated probability set makes difficult the use of classic tools, such as the dominated convergence theorem and Komlos separation lemma, to analyze the nonlinear expectation associated to the probability set. This project aims to develop new methods to handle the much more complicated probabilistic situations under model uncertainty. These are expected to be useful also in the more general subjects of stochastic control and optimization.