Probabilistic Approach to Rough PDEs: Applications to Finance and Control

本研究项目涉及不确定条件下最优决策的数学理论。将特别关注量化金融中出现的问题，其中一个经典问题是最大化投资者的未来收益或将他们未来的风险降至最低。我们将进一步发展支持此类问题建模和研究的概率理论。将特别关注除随机性外，模型还必须反映系统中可能发生的某些特定不连续的情况。一个例子是对一家公司的最佳投资，该公司可能会面临破产程序，或者更广泛地说，可能会发生重大变化。除了发展数学理论外，我们还致力于加深对这类系统的数值计算的理解，以便该项目的结果将直接惠及金融工程界和学术界。此外，拟议的研究活动将包括对本科生和研究生的指导，以及通过在科学期刊上发表演讲和发表文章来传播科学。众所周知，许多最优随机控制问题的值和最优策略都可以用(正)倒向随机微分方程来刻画。然而，这些方程因适定性而对系数施加了相当强的正则性条件，严重限制了用这些概率方法解决控制问题的领域。在第一部分，本项目的目的是发展一套新的思想，用于研究具有粗系数的倒向随机微分方程解的适定性和正则性，以及与之相关的二阶偏微分方程解的适定性和正则性。我们设想的主要方法将充分利用Malliavin微积分理论中的强紧性准则。然后，我们将考虑一些应用，包括在粗糙偏微分方程组、定量金融(投资组合优化问题)和最优运输方面的应用。将我们的理论结果与新开发的用于倒向随机微分方程近似的深度学习方法结合在一起，将进一步允许对在一些应用领域构建的解进行数值模拟。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

This research project concerns the mathematical theory of optimal decision making under uncertainty. A particular focus will be put on problems arising in quantitative finance, where a classical issue is to maximize an investor’s future gain or to minimize their future risk. We will further develop the probabilistic theory underpinning the modeling and investigation of such problems. A particular focus will be put on situations where, in addition to randomness, models have to reflect some particular discontinuities that can occur in the system. One example is that of optimal investment in a company subject to possible bankruptcy proceedings or, more generally, subject to possible drastic changes. In addition to developing the mathematical theory, we also aim to further the understanding of the numerical computation of such systems so that the outcomes of the project will directly benefit the financial engineering and academic communities. Moreover, the proposed research activity will include mentorship of undergraduate as well as graduate students, and scientific dissemination through presentations and publications in scientific journals. It is well-known that the value and optimal strategies of many optimal stochastic control problems can be characterized by (forward) backward stochastic differential equations. However, the rather strong regularity conditions imposed on coefficient of these equations for their well-posedness severely restrict the realm of control problems that can be tackled with these probabilistic methods. The goal of this project is, in the first part, to develop a set of new ideas that can be used to study well-posedness and regularity of solutions of backward stochastic differential equations with rough coefficients and the associated second order partial differential equations. The main approach we envision will make ample use of strong compactness criteria from the theory of Malliavin calculus. Then, we will consider a number of applications, including application to rough partial differential equations, quantitative finance (portfolio optimization problems) and optimal transport. Putting our theoretical results together with newly developed deep learning approaches to the approximation of backward stochastic differential equations will further allow to numerically simulate the solutions constructed in some of the application areas.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.