Neural networks for stochastic control and decision making in mathematical finance

Abstract

As the title suggests, this thesis is devoted to machine learning algorithms for stochastic control with applications in mathematical finance.

In the first two chapters, we utilize neural networks to find the optimal exercise strategies of high-dimensional American options. Moreover, we compute option values along stochastic trajectories of the underlying assets, which is crucial in modern risk management.

In Chapter 3, we approximate strongly coupled FBSDEs with a sequence of neural networks. More specifically, we are concerend with high-dimensional FBSDEs stemming from stochastic optimal control problems. These equations are inherently difficult to approximate due the coupling, and most classical methods as well as modern day machine learning algorithms break down. Our approach is novel in the sense that it combines mathematical structure from the initial control problem with the associate FBSDE which is shown to be both robust and accurate for a wide range of problems.

In the final chapter, we consider time-inconsistent portfolio optimization problems. In general, for time-inconsistent problems the dynamic programming principle does not hold and in turn, the global problem cannot be divided into local sub-problems. Instead, we take on a fully data driven neural network-based approach to approximate the optimal allocation strategy directly. Beyond the classical instruments stocks and bonds, we also allow the investor to buy and sell options. In this way, we increase the flexibility in shaping the terminal wealth distribution.