Optimal randomized multilevel Monte Carlo for repeatedly nested expectations

Yasa Syed and Guanyang Wang

Papers from arXiv.org

Abstract: The estimation of repeatedly nested expectations is a challenging task that arises in many real-world systems. However, existing methods generally suffer from high computational costs when the number of nestings becomes large. Fix any non-negative integer $D$ for the total number of nestings. Standard Monte Carlo methods typically cost at least $\mathcal{O}(\varepsilon^{-(2+D)})$ and sometimes $\mathcal{O}(\varepsilon^{-2(1+D)})$ to obtain an estimator up to $\varepsilon$-error. More advanced methods, such as multilevel Monte Carlo, currently only exist for $D = 1$. In this paper, we propose a novel Monte Carlo estimator called $\mathsf{READ}$, which stands for "Recursive Estimator for Arbitrary Depth.'' Our estimator has an optimal computational cost of $\mathcal{O}(\varepsilon^{-2})$ for every fixed $D$ under suitable assumptions, and a nearly optimal computational cost of $\mathcal{O}(\varepsilon^{-2(1 + \delta)})$ for any $0

Date: 2023-01, Revised 2023-05
New Economics Papers: this item is included in nep-cmp
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