A weighted multilevel Monte Carlo method

ArXiv ID: 2405.03453 “View on arXiv”

Authors: Unknown

Abstract

The Multilevel Monte Carlo (MLMC) method has been applied successfully in a wide range of settings since its first introduction by Giles (2008). When using only two levels, the method can be viewed as a kind of control-variate approach to reduce variance, as earlier proposed by Kebaier (2005). We introduce a generalization of the MLMC formulation by extending this control variate approach to any number of levels and deriving a recursive formula for computing the weights associated with the control variates and the optimal numbers of samples at the various levels. We also show how the generalisation can also be applied to the \emph{“multi-index”} MLMC method of Haji-Ali, Nobile, Tempone (2015), at the cost of solving a $(2^d-1)$-dimensional minimisation problem at each node when $d$ index dimensions are used. The comparative performance of the weighted MLMC method is illustrated in a range of numerical settings. While the addition of weights does not change the \emph{“asymptotic”} complexity of the method, the results show that significant efficiency improvements over the standard MLMC formulation are possible, particularly when the coarse level approximations are poorly correlated.

Keywords: Monte Carlo methods, Multilevel Monte Carlo, Control variates, Variance reduction, Derivatives pricing

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 4.0/10
  • Quadrant: Lab Rats
  • Why: The paper introduces a novel mathematical generalization of Multilevel Monte Carlo using recursive control variate weighting and solving high-dimensional minimization problems, featuring advanced probability theory and complex derivations. While it includes numerical examples, it lacks real financial data, backtests, or performance metrics, focusing instead on mathematical theory and synthetic test settings.
  flowchart TD
    A["Research Goal:<br/>Generalize MLMC & Derive Optimal Weights"] --> B["Methodology:<br/>Recursive Weight Formulation"]
    B --> C["Data Inputs:<br/>Hierarchical Discretization Levels"]
    C --> D["Computational Process:<br/>Weighted Control Variate Estimation"]
    D --> E{"Outcome:<br/>Complexity Analysis"}
    E -->|Asymptotic| F["Complexity Unchanged<br/>(Same as Standard MLMC)"]
    E -->|Empirical| G["Significant Efficiency Gain<br/>(Especially for Low Correlation)"]