Robust Methods in Mathematical Finance

长期以来，随机模型一直被成功地应用于描述金融市场、做出投资决策、为金融衍生品定价和对冲以及管理风险。但这导致了对特定型号的过度依赖，并忽视了型号错误说明的可能性。虽然在大多数工程应用中，稳健方法已经有了很长的历史，但数学金融中的标准方法仍然是用单一的概率度量来描述所有潜在的不确定性。然而，在许多现实生活中，不可能确定未来事件的准确概率，如果现实没有完全像模型预测的那样展开，对特定假设敏感的财务决策规则可能会导致灾难性的结果。这个项目的目标是开发考虑到极端事件的财务决策方法，并对模型错误指定具有健壮性。预计结果将导致在投资、资产定价、对冲和风险管理方面采取强有力的方法。研究生被包括在该项目的工作中。该项目旨在创建稳健的方法，可用于解决数学金融中的相关问题。研究了离散时间模型和连续时间模型。该计划是建立非线性Riesz表示结果，这些结果可用于开发资产定价基本定理的稳健版本，以及相应的定价对偶公式、定价和对冲期权的稳健方法，以及处理模型不确定性下的最优资产配置问题的方法。需要解决的问题是数学金融学、投资理论和金融经济学的交叉点。研究方法运用了概率论、随机分析、泛函分析、决策论等方法。

Stochastic models have long been successfully applied to describe financial markets, make investment decisions, price and hedge financial derivatives, and manage risks. But this has led to an over-reliance on particular models and a disregard of the possibility of model misspecification. While in most engineering applications robust methods have a long history, the standard approach in mathematical finance is still to describe all underlying uncertainty with a single probability measure. However, in many real-life situations it is not possible to determine the precise probabilities of future events, and financial decision rules that are sensitive to particular assumptions can lead to disastrous outcomes if reality does not exactly unfold as predicted by the model. The goal of this project is to develop methods of financial decision-making that take into account extreme events and are robust with respect to model misspecification. The outcomes are expected to lead to robust approaches to investing, asset pricing, hedging, and risk management. Graduate students are included in the work of the project. The project aims to create robust methods that can be used to address relevant problems in mathematical finance. Discrete-time as well as continuous-time models are studied. The plan is to establish non-linear Riesz representation results that can be used to develop robust versions of the fundamental theorem of asset pricing together with corresponding pricing duality formulas, robust methods for pricing and hedging options, as well as approaches to treat optimal asset allocation problems under model uncertainty. The problems to be addressed lie at the intersection of mathematical finance, investment theory, and financial economics. Methods from probability theory, stochastic analysis, functional analysis, and decision theory are used.