Residual Monte Carlo: A Unifying Framework for Variance Reduction

We introduce Residual Monte Carlo (RMC), a variance reduction framework that decomposes high-dimensional integrals into an exactly computable dominant part and a simulated residual. The decomposition is driven by the eigenvalue spectrum of the correlation structure: the \(K\) dominant modes — mutually independent projections onto the leading eigenvectors — are integrated exactly via deterministic quadrature or the COS spectral method, while the residual modes (eigenvalues \(\lambda_{K+1}, \lambda_{K+2}, \ldots\)) are estimated by importance sampling or quasi-Monte Carlo.

We establish three main results. First, the Variance Annihilation Bound (Theorem 1): the RMC estimator variance satisfies \[ \text{Var}(\hat{\mu}_{\text{RMC}}) \leq \frac{\sum_{k > K} \lambda_k}{\text{tr}(C)} \cdot \frac{C(\ell) \cdot \rho^{-2\ell}}{N_{\text{res}}} \] where \(\rho > 1\) is the analyticity radius, \(\ell\) is the loss threshold, and \(N_{\text{res}}\) is the number of residual samples. In the finite Latent case (\(\lambda_{K+1} = 0\)), the variance is exactly zero. Second, the Unification Theorem (Theorem 2): each classical variance reduction family — importance sampling, control variates, stratified sampling, quasi-Monte Carlo, antithetic variates, and conditional Monte Carlo — is recovered as a special case or approximation of RMC at a specific truncation level and residual strategy. Third, the Sobol Acceleration Theorem (Theorem 3): when Sobol sequences replace random sampling in the residual phase, the RMC error rate improves from \(O(N^{-1/2})\) to \(O(N^{-1}(\log N)^{d-K})\), with the effective QMC dimension reduced from \(d\) to \(d - K\).

The method applies to any problem with a spectral gap (\(\lambda_1 / \lambda_2 > 1\)) and analytic characteristic functions (\(\rho > 1\)). We demonstrate RMC on deep-tail risk estimation (\(10^{-8}\) level), multi-asset barrier options, and credit portfolio losses, achieving variance reductions of \(10^4\)–\(10^7\) over naive Monte Carlo.

Keywords: Monte Carlo, variance reduction, eigenvalue decomposition, quasi-Monte Carlo, Sobol sequences, importance sampling, Rao-Blackwellization, spectral methods

MSC 2020: 65C05, 65D30, 60F10, 91G60