Conditional Quasi-Monte Carlo with Constrained Active Subspaces

Conditional Monte Carlo or pre-integration is a useful tool for reducing variance and improving regularity of integrands when applying Monte Carlo and quasi-Monte Carlo (QMC) methods. To choose the variable to pre-integrate with, one need to consider both the variable importance and the tractability of the conditional expectation. For integrals over a Gaussian distribution, one can pre-integrate over any linear combination of variables. Liu and Owen (2022) propose to choose the linear combination based on an active subspace decomposition of the integrand. However, pre-integrating over such selected direction might be intractable. In this work, we address this issue by finding the active subspaces subject to the constraints such that pre-integration can be easily carried out. The proposed method is applied to some examples in derivative pricing under stochastic volatility models and is shown to outperform previous methods.