Control variates for dry friction under colored noise: variance rates for non-smooth observables
Abstract
A control variate method was introduced by Garnier and Mertz (2022) for multivalued stochastic differential equations driven by colored noise. For smooth observables, the asymptotic variance is O(ε2), where ε2 is the correlation time of the noise. However, for non-smooth reliability observables, such as threshold indicators and overshoot severities, rate estimates have not been established in general. In this paper, rate estimates are derived for the inertial dry-friction inclusion driven by colored noise in the three regimes whose white-noise propagators are known. The asymptotic variance of the resulting control variate estimators is O(ε2α) for α-Hölder observables with α∈(0,1], O(ε) for observables of bounded variation satisfying a one-sided continuity condition, O(ε1+α) for observables that are both α-Hölder and of bounded variation, and O(εmin{2,1+2α}) for observables with finitely many α-cusps, where α∈ (0, 1] is the cusp exponent. The last three bounds hold up to logarithmic factors. Bounds on the bias of the white-noise approximation and non-asymptotic sample-size estimates are also derived, and the comparison principle is shown to be preserved under coupled time discretization. Numerical experiments in the three regimes support the predicted rates.
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BibTeX
@misc{guo2026control,
title = {{Control variates for dry friction under colored noise: variance rates for non-smooth observables}},
author = {Lihong Guo},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.27765},
eprint = {2610.27765},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.27765},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Control variates for dry friction under colored noise: variance rates for non-smooth observables
AU - Lihong Guo
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.27765
UR - https://arxiv.org/abs/2610.27765
AB - A control variate method was introduced by Garnier and Mertz (2022) for multivalued stochastic differential equations driven by colored noise. For smooth observables, the asymptotic variance is O(ε^2), where ε^2 is the correlation time of the noise. However, for non-smooth reliability observables, such as threshold indicators and overshoot severities, rate estimates have not been established in general. In this paper, rate estimates are derived for the inertial dry-friction inclusion driven by colored noise in the three regimes whose white-noise propagators are known. The asymptotic variance of the resulting control variate estimators is O(ε^2α) for α-Hölder observables with α∈(0,1], O(ε) for observables of bounded variation satisfying a one-sided continuity condition, O(ε^1+α) for observables that are both α-Hölder and of bounded variation, and O(ε^min{2,1+2α}) for observables with finitely many α-cusps, where α∈ (0, 1] is the cusp exponent. The last three bounds hold up to logarithmic factors. Bounds on the bias of the white-noise approximation and non-asymptotic sample-size estimates are also derived, and the comparison principle is shown to be preserved under coupled time discretization. Numerical experiments in the three regimes support the predicted rates.
ER -