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

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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.