Probabilistic Approach to Rough PDEs: Applications to Finance and Control

点击翻译按钮获取中文摘要

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.