Explainable Artificial Intelligence for Financial Integral Equations: A Fixed-Point Neural Operator Approach

Abstract

The explainable artificial intelligence is used to analyze the stochastic Fredholm integral equations (SFIEs) and stochastic deep neural networks (SDNNs). The neural operator-based stochastic fixed point framework is used to develop SDNNs. The solution of an SFIE is obtained through successive applications of an integral operator, and this iterative structure naturally resembles the layered architecture of a neural network. The associated nonlinear versions of SFIE and SDNN are discussed. The SFIE and SDNN are used to solve the Black-Scholes equation, contagion dynamics of financial networks, and the Merten jump diffusion equation. It is observed that the results obtained through SFIE and SDNN for all the applications agree well.