Functional Calculus for Pathwise Hedging

潜在概率动态的不确定性是金融领域的一个关键问题。的确，在这种情况下，每个模型充其量只是对现实的一个有用但粗略的近似。因此，辨别哪些结果微妙地依赖于所选择的模型假设，哪些结果是稳健的，因为它们可以从广泛的定性性质中推断出来，这一点至关重要。因此，模型不确定性的分析以及如何通过金融衍生工具的稳健定价和套期保值来考虑模型不确定性是当前研究的重点方向。从数学的角度来看，这自然会导致关于随机微积分的哪些部分可以以纯粹的路径方式发展的深刻问题。本研究项目将在随机分析及其金融应用的交叉领域做出深远的贡献。这方面的关键工具是“泛函演算”，它描述泛函在一般的、路径相关的随机系统上的作用。在金融环境中，这允许将“超级对冲策略”（完全对冲给定金融风险）与路径相关的最优性方程联系起来。这些反过来又导致在一些具体的例子中显式的解决方案，并且通常打开了有效的数值方法的部署的大门。本项目在许多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与决定其收益的基础资产对冲，而且还通过不断调整更简单的衍生品头寸来对冲。这种风险管理策略在实践中经常使用，但其基本理论尚未得到很好的理解——本项目将使用函数演算来填补文献中的空白。除了这些金融应用之外，本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.