Neural variance reduction for stochastic differential equations

TL;DR

This paper introduces neural SDEs with neural network-based control variates to effectively reduce variance in Monte Carlo simulations for complex stochastic differential equations, including those driven by Lévy processes.

Contribution

It presents a novel approach combining neural networks with SDE control variates for variance reduction, applicable to a broad class of stochastic processes including infinite activity Lévy processes.

Findings

Neural control variates significantly reduce variance in Monte Carlo simulations.

The method is effective for SDEs driven by Brownian motion and Lévy processes.

Numerical examples demonstrate improved efficiency in option pricing applications.

Abstract

Variance reduction techniques are of crucial importance for the efficiency of Monte Carlo simulations in finance applications. We propose the use of neural SDEs, with control variates parameterized by neural networks, in order to learn approximately optimal control variates and hence reduce variance as trajectories of the SDEs are being simulated. We consider SDEs driven by Brownian motion and, more generally, by L\'{e}vy processes including those with infinite activity. For the latter case, we prove optimality conditions for the variance reduction. Several numerical examples from option pricing are presented.