Deep Kusuoka Approximation: High-Order Spatial Approximation for Solving High-Dimensional Kolmogorov Equations and Its Application to Finance

The paper introduces a new deep learning-based high-order spatial approximation for a solution of a high-dimensional Kolmogorov equation where the initial condition is only assumed to be a continuous function and the condition on the vector fields associated with the differential operator is very general, i.e. weaker than Hörmander’s hypoelliptic condition. In particular, the deep learning-based method is constructed based on the Kusuoka approximation. Numerical results for high-dimensional partial differential equations up to 500-dimension cases appearing in option pricing problems show the validity of the method. As an application, a computation scheme for the delta is shown using “deep” numerical differentiation.