Smooth Nested Simulation: Bridging Cubic and Square Root Convergence Rates in High Dimensions

TL;DR

This paper introduces a kernel ridge regression-based nested simulation method that leverages smoothness to improve convergence rates in high-dimensional settings, effectively bridging the gap between cubic and square root rates.

Contribution

It proposes a novel approach that exploits smoothness of the conditional expectation to mitigate the curse of dimensionality in nested simulation.

Findings

Effective reduction of convergence rate from cubic to square root with sufficient smoothness.

Demonstrated improvements in portfolio risk management and uncertainty quantification tasks.

Theoretical analysis confirms the method's asymptotic efficiency.

Abstract

Nested simulation concerns estimating functionals of a conditional expectation via simulation. In this paper, we propose a new method based on kernel ridge regression to exploit the smoothness of the conditional expectation as a function of the multidimensional conditioning variable. Asymptotic analysis shows that the proposed method can effectively alleviate the curse of dimensionality on the convergence rate as the simulation budget increases, provided that the conditional expectation is sufficiently smooth. The smoothness bridges the gap between the cubic root convergence rate (that is, the optimal rate for the standard nested simulation) and the square root convergence rate (that is, the canonical rate for the standard Monte Carlo simulation). We demonstrate the performance of the proposed method via numerical examples from portfolio risk management and input uncertainty quantification.