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

TL;DR

This paper introduces a neural operator-based fixed point framework for solving stochastic integral equations in finance, providing explainability and effective solutions for complex financial models.

Contribution

It develops a novel neural operator approach for stochastic integral equations, enhancing explainability and applicability in financial modeling.

Findings

SFIE and SDNN solutions agree well across applications

The framework effectively solves Black-Scholes, contagion, and jump diffusion equations

The iterative neural network structure naturally models integral equations

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.