Stochastic viscosity approximations of Hamilton–Jacobi equations and variance reduction

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