Stochastic viscosity approximations of Hamilton-Jacobi equations and variance reduction

TL;DR

This paper introduces a new framework using stochastic viscosity approximations for Hamilton-Jacobi equations to quantitatively assess and reduce variance in high-dimensional Monte Carlo simulations of diffusions, especially in low noise regimes.

Contribution

It proposes the concept of k-stochastic viscosity approximation (SVA) for HJB equations, providing a criterion to evaluate variance reduction effectiveness and demonstrating bounded variance schemes as noise diminishes.

Findings

1-SVA schemes have bounded variance as noise approaches zero.

Higher-order SVAs achieve lower variance at the log-scale.

Numerical examples confirm theoretical variance reduction results.

Abstract

We consider the computation of free energy-like quantities for diffusions in high dimension, when resorting to Monte Carlo simulation is necessary. Such stochastic computations typically suffer from high variance, in particular in a low noise regime, because the expectation is dominated by rare trajectories for which the observable reaches large values. Although importance sampling, or tilting of trajectories, is now a standard technique for reducing the variance of such estimators, quantitative criteria for proving that a given control reduces variance are scarce, and often do not apply to practical situations. The goal of this work is to provide a quantitative criterion for assessing whether a given bias reduces variance, and at which order. We rely for this on a recently introduced notion of stochastic solution for Hamilton-Jacobi-Bellman (HJB) equations. Based on this tool, we introduce the notion of k-stochastic viscosity approximation (SVA) of a HJB equation. We next prove that such approximate solutions are associated with estimators having a relative variance of order k-1 at log-scale. In particular, a sampling scheme built from a 1-SVA has bounded variance as noise goes to zero. Finally, in order to show that our definition is relevant, we provide examples of stochastic viscosity approximations of order one and two, with a numerical illustration confirming our theoretical findings.