| Abstract: |
| We develop a generic convolutional neural network (CNN) and machine learning based numerical scheme to simulate the 2-tuple adapted strong solution to a unified backward stochastic partial differential equation (B-SPDE) driven by Brownian motion with application to a strongly nonlinear case in finance. The dynamics of the scheme is modeled by a CNN through conditional expectation projection and machine learning. It consists of two convolution parts: W layers of backward networks and L layers of reinforcement iterations. Furthermore, it is a completely discrete and iterative algorithm in terms of both time and space with mean-square error estimation and almost sure convergence supported by both theoretical proof and numerical examples. In doing so, we need to prove the unique existence of the 2-tuple adapted strong solution to the system under both conventional and Malliavin derivatives with general local Lipschitz and linear growth conditions. |
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