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arxiv: 2304.07797 · v1 · submitted 2023-04-16 · 🧮 math.ST · math.OC· math.PR· stat.TH

Optimal distributions for randomized unbiased estimators with an infinite horizon and an adaptive algorithm

Chao Zheng , Jiangtao Pan , Qun Wang This is my paper

Pith reviewed 2026-05-24 09:20 UTC · model grok-4.3

classification 🧮 math.ST math.OCmath.PRstat.TH
keywords randomized unbiased estimatorsinfinite horizonoptimal distributionsadaptive algorithmstochastic differential equationsRhee-Glynn estimatorsvariance reduction
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The pith

Under mild assumptions the optimal distributions for infinite-horizon randomized unbiased estimators admit a simple closed-form representation.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that randomized unbiased estimators for expectations of path functionals of stochastic differential equations can be made more efficient when the sampling distributions are chosen optimally. For the infinite-horizon case, where earlier methods lacked practical algorithms, the authors prove that a simple representation of those optimal distributions exists once mild conditions are met. They then construct an adaptive algorithm that finds the distributions using only modest prior computation. The result matters because these estimators are already known to be highly efficient when the distributions are right; the new representation removes the main barrier to using them at long time horizons.

Core claim

Based on the method of Cui et al., the authors prove that under mild assumptions there is a simple representation of the optimal distributions for randomized unbiased estimators with an infinite horizon. They then develop an adaptive algorithm to compute the optimal distributions, which requires only a small amount of computational time in prior estimation, and illustrate its efficiency with numerical results.

What carries the argument

The simple representation of the optimal distributions, obtained by applying the Cui et al. method to the infinite-horizon setting, which directly supplies the parameters needed by the adaptive algorithm.

If this is right

  • The adaptive algorithm produces the optimal distributions with far less prior computation than exhaustive search methods.
  • Numerical experiments confirm that the computed distributions yield lower variance estimators than non-optimal choices.
  • The representation extends the practical reach of Rhee-Glynn-style unbiased estimators to problems with unbounded time horizons.
  • Once the representation is known, the same adaptive procedure can be reused for different payoff functionals without restarting the entire optimization.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The representation may allow closed-form or semi-closed-form solutions for specific classes of SDEs that satisfy the mild assumptions.
  • The adaptive algorithm could be combined with existing variance-reduction techniques such as control variates to achieve further efficiency gains.
  • If the mild assumptions fail in a given application, the representation supplies a natural starting point for deriving corrections or bounds.

Load-bearing premise

The mild assumptions required to obtain the simple representation from the Cui et al. method are satisfied.

What would settle it

Apply the representation to a concrete infinite-horizon SDE example where the mild assumptions can be verified directly; if the resulting distribution is not optimal, the claim is false.

read the original abstract

The randomized unbiased estimators of Rhee and Glynn (Operations Research:63(5), 1026-1043, 2015) can be highly efficient at approximating expectations of path functionals associated with stochastic differential equations (SDEs). However, there is a lack of algorithms for calculating the optimal distributions with an infinite horizon. In this article, based on the method of Cui et.al. (Operations Research Letters: 477-484, 2021), we prove that, under mild assumptions, there is a simple representation of the optimal distributions. Then, we develop an adaptive algorithm to compute the optimal distributions with an infinite horizon, which requires only a small amount of computational time in prior estimation. Finally, we provide numerical results to illustrate the efficiency of our adaptive algorithm.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The manuscript claims that, based on the method of Cui et al. (2021), there exists a simple representation of the optimal distributions for randomized unbiased estimators with an infinite horizon under mild assumptions. It develops an adaptive algorithm to compute these distributions that requires only a small amount of prior estimation time, and provides numerical results to demonstrate the algorithm's efficiency for approximating expectations of SDE path functionals.

Significance. If the representation holds and transfers validly, the result would address a gap in algorithms for optimal distributions in the infinite-horizon setting of Rhee-Glynn randomized unbiased estimators, enabling more efficient Monte Carlo approximations for SDE expectations. The adaptive algorithm's low prior-computation requirement would be a practical contribution if the underlying representation is rigorously justified.

major comments (1)
  1. [§3] §3: The proof of the simple representation invokes the Cui et al. (2021) method but does not explicitly verify or restate the transfer of the required conditions on the cost/variance functionals and on existence/uniqueness to the infinite-horizon randomized estimator setting; the 'mild assumptions' may therefore need to be augmented by additional regularity (e.g., uniform integrability over infinite paths) that is not shown to hold.
minor comments (1)
  1. [Abstract] Abstract: the citation is written 'Cui et.al.'; standard formatting is 'Cui et al.'

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for identifying an important point regarding the rigor of the proof in §3. We address the major comment below and will incorporate the necessary clarifications in a revised version.

read point-by-point responses

  1. Referee: [§3] §3: The proof of the simple representation invokes the Cui et al. (2021) method but does not explicitly verify or restate the transfer of the required conditions on the cost/variance functionals and on existence/uniqueness to the infinite-horizon randomized estimator setting; the 'mild assumptions' may therefore need to be augmented by additional regularity (e.g., uniform integrability over infinite paths) that is not shown to hold.

    Authors: We agree that an explicit verification of the transfer of conditions from Cui et al. (2021) to the infinite-horizon setting would strengthen the manuscript. In the revision, we will add a new subsection (or appendix) to §3 that restates the relevant conditions on the cost and variance functionals, confirms existence and uniqueness, and provides a short argument establishing uniform integrability over infinite paths under the paper's mild assumptions. This will make the applicability of the representation fully rigorous without changing the main results or assumptions. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation extends external Cui et al. 2021 method

full rationale

The paper's central result (simple representation of optimal distributions) is explicitly derived from the method in the external reference Cui et al. (Operations Research Letters, 2021). No self-citations are load-bearing, no fitted parameters are renamed as predictions, and no equations reduce by construction to the paper's own inputs. The derivation chain is self-contained against the cited external benchmark under the stated mild assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the mild assumptions needed for the representation (domain assumption) and on the prior method of Cui et al. 2021 (standard_math). No free parameters or invented entities are stated in the abstract.

axioms (1)
  • domain assumption Mild assumptions sufficient for the simple representation of optimal distributions
    Invoked to prove the representation (abstract and reference to Cui et al. method)

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Reference graph

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