Neural stochastic Volterra equations: learning path-dependent dynamics

TL;DR

This paper introduces neural stochastic Volterra equations as a new architecture for modeling complex, path-dependent stochastic systems, providing theoretical insights and demonstrating superior performance in diverse numerical experiments.

Contribution

It generalizes neural stochastic differential equations to include memory effects, offering a novel physics-inspired framework with theoretical foundations.

Findings

Neural SVEs outperform neural SDEs and DeepONets in experiments.

The approach effectively models systems with memory and irregular behavior.

Numerical results validate the theoretical advantages of neural SVEs.

Abstract

Stochastic Volterra equations (SVEs) serve as mathematical models for the time evolutions of random systems with memory effects and irregular behaviour. We introduce neural stochastic Volterra equations as a physics-inspired architecture, generalizing the class of neural stochastic differential equations, and provide some theoretical foundation. Numerical experiments on various SVEs, like the disturbed pendulum equation, the generalized Ornstein--Uhlenbeck process, the rough Heston model and a monetary reserve dynamics, are presented, comparing the performance of neural SVEs, neural SDEs and Deep Operator Networks (DeepONets).