Introduction to Solving Quant Finance Problems with Time-Stepped FBSDE and Deep Learning

In this introductory paper, we discuss how quantitative finance problems under some common risk factor dynamics for some common instruments and approaches can be formulated as time-continuous or time-discrete forward-backward stochastic differential equations (FBSDE) final-value or control problems, how these final value problems can be turned into control problems, how time-continuous problems can be turned into time-discrete problems, and how the forward and backward stochastic differential equations (SDE) can be time-stepped. We obtain both forward and backward time-stepped time-discrete stochastic control problems (where forward and backward indicate in which direction the $Y$ SDE is time-stepped) that we will solve with optimization approaches using deep neural networks for the controls and stochastic gradient and other deep learning methods for the actual optimization/learning. We close with examples for the forward and backward methods for an European option pricing problem. Several methods and approaches are new.