Functional Calculus for Pathwise Hedging

潜在概率动态的不确定性是金融学中的一个关键问题。事实上，在这种背景下，每个模型充其量都是对现实的有用但粗略的近似。因此，关键是要辨别哪些结果微妙地依赖于所选的模型假设，哪些结果是稳健的，因为它们可以从广泛的定性属性中推导出来。因此，模型不确定性的分析以及如何通过金融衍生产品的稳健定价和套期保值将其考虑在内是当前研究的重点方向。从数学的角度来看，这自然会引发深刻的问题，即随机微积分的哪些部分可以以纯粹的路径方式发展。本研究项目将对随机分析及其金融应用的交叉点做出深刻的贡献。这里的关键工具是“泛函演算”，它描述泛函在一般路径依赖的随机系统上的作用。在金融背景下，这允许将“超级对冲策略”(完全对冲给定的金融风险)与路径依赖的最优化方程联系起来。这些反过来又在一些具体的例子中导致了显式的解，并且通常为部署有效的数值方法打开了大门。本项目在一些实际重要的情况下探讨了这种方法，例如，复杂的金融衍生品不仅用决定其收益的基础资产进行对冲，而且还通过不断调整较简单衍生品的头寸来进行对冲。这样的风险管理策略在实践中经常被使用，但基本的理论并不被很好地理解-这是文献中的一个空白，将在这个项目中使用函数演算来填补。除了这些金融应用外，拟议的研究还将以一些基本的方式进一步发展泛函演算的一般理论，例如，不随时间连续演化的泛函。

Uncertainty about the underlying probabilistic dynamics is a key problem in finance. Indeed, in this context, every model is at best a useful but rough approximation of reality. It is therefore crucial to discern which results depend delicately on the chosen model assumptions, and which ones are robust, in that they can be deduced from broad qualitative properties. Accordingly, the analysis of model uncertainty and how to take it into account via the robust pricing and hedging of financial derivatives are key directions of current research. From a mathematical perspective, this naturally leads to deep questions about what parts of stochastic calculus can be developed in a purely pathwise manner. The present research project will make profound contributions at this intersection of stochastic analysis and its financial applications. The key tool in this context is "functional calculus", which describes the actions of functionals on general, path-dependent random systems. In a financial context, this allows to link "superhedging strategies" (that completely hedge a given financial risk) to path-dependent optimality equations. These in turn lead to explicit solutions in some concrete examples and generally open the door to the deployment of efficient numerical methods. The present project explores this approach in a number of practically important settings, e.g., the case where a complex financial derivative is not only hedged with the underlying asset that determines its payoff, but also by continuously readjusting a position in simpler derivatives. Such risk management strategies are routinely used in practice, but the underlying theory is not well understood - a gap in the literature that will be filled in this project using functional calculus. In addition to these financial applications, the proposed research will also further develop the general theory of functional calculus in a number of fundamental ways, e.g., to functionals that do not evolve continuously in time.