Some Mathematical Finance Problems Under Model Uncertainty

最近，模型不确定性下的金融问题研究重新引起了人们的关注。从实用的角度来看，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；考虑所有与市场数据兼容的概率模型是更合理的。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将如何演变有不同的信念或预测，对相同的刺激有不同的反应。在本研究项目中，研究者研究了在模型不确定性下与博弈论中的微分均衡和纳什均衡、分位数对冲和带有交易成本的衍生品定价相关的几个随机优化问题。这些研究旨在阐明模型不确定性的某些方面，不仅可以应用于金融市场，也可以应用于其他动态系统。数学上，模型的不确定性通常由一组非支配的相互奇异概率表示（在一种概率下可以忽略不计的集合在另一种概率下可能具有正质量）。研究了在非支配概率集中不确定评价准则变化时的4个随机优化问题：(1)确定鲁棒Dynkin对策是否有值并允许最优三元组；(2)确定多参与者之间的稳健非零和博弈是否具有纳什均衡；(3)寻找鲁棒分位数对冲问题的最优策略；(4)寻找具有交易成本的资产定价基本定理的连续时间鲁棒形式的无套利条件。由于在非支配概率集中缺乏参考概率，使得难以使用经典工具，如支配收敛定理和Komlos分离引理来分析与概率集相关的非线性期望。本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。这些也有望在更一般的随机控制和优化主题中发挥作用。

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.