Convergence proofs and strong error bounds for forward-backward stochastic differential equations using neural network simulations

TL;DR

This paper establishes convergence proofs and provides strong error bounds for neural network-based simulations of forward-backward stochastic differential equations, connecting them with PDE solutions via the Feynman-Kac theorem.

Contribution

It introduces a novel numerical analysis framework that combines multilevel Monte Carlo methods with neural network approximations for stochastic differential equations.

Findings

Bound the strong error in terms of discretisation and neural network approximation errors.

Identify limitations of the current loss function used in neural network approximations.

Propose frameworks to exploit variance structures in multilevel Monte Carlo estimators.

Abstract

We introduce forward-backward stochastic differential equations, highlighting the connection between solutions of these and solutions of partial differential equations, related by the Feynman-Kac theorem. We review the technique of approximating solutions to high dimensional partial differential equations using neural networks, and similarly approximating solutions of stochastic differential equations using multilevel Monte Carlo. Connecting the multilevel Monte Carlo method with the neural network framework using the setup established by E et al. and Raissi, we provide novel numerical analyses to produce strong error bounds for the specific framework of Raissi. Our results bound the overall strong error in terms of the maximum of the discretisation error and the neural network's approximation error. Our analyses are necessary for applications of multilevel Monte Carlo, for which we propose suitable frameworks to exploit the variance structures of the multilevel estimators we elucidate. Also, focusing on the loss function advocated by Raissi, we expose the limitations of this, highlighting and quantifying its bias and variance. Lastly, we propose various avenues of further research which we anticipate should offer significant insight and speed improvements.