Probabilistic Approach to Rough PDEs: Applications to Finance and Control

本课题研究不确定条件下最优决策的数学理论。一个特别的重点将放在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最小化他们的未来风险。我们将进一步发展支持这类问题的建模和调查的概率论。除了随机性之外，模型还必须反映系统中可能出现的一些特定的不连续性，将特别关注这些情况。一个例子是对一家可能面临破产程序的公司的最优投资，或者更一般地说，对一家可能面临剧烈变化的公司的最优投资。除了发展数学理论外，我们还旨在进一步了解这些系统的数值计算，以便项目的成果将直接造福于金融工程和学术界。此外，拟议的研究活动将包括指导本科生和研究生，并通过在科学期刊上发表演讲和出版物进行科学传播。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）后向随机微分方程来表征。然而，这些方程的系数具有很强的正则性条件，严重限制了用这些概率方法解决控制问题的范围。本项目的目标是，在第一部分中，发展一套新的思想，可用于研究具有粗糙系数的倒向随机微分方程及其相关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，包括应用于粗糙偏微分方程，定量金融（投资组合优化问题）和最优运输。将我们的理论结果与新开发的深度学习方法结合起来，以近似后向随机微分方程，将进一步允许数值模拟在一些应用领域中构建的解决方案。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值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.