Control variates for dry friction under colored noise: variance rates for non-smooth observables

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 $O({\epsilon }^2)$, where ${\epsilon }^2$ 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 $O({\epsilon }^{2\alpha })$ for $\alpha$-Hölder observables with $\alpha \in (0,1]$, $O(\epsilon )$ for observables of bounded variation satisfying a one-sided continuity condition, $O({\epsilon }^{1+\alpha })$ for observables that are both $\alpha$-Hölder and of bounded variation, and $O({\epsilon }^{\min \{2,1+2\alpha \}})$ for observables with finitely many $\alpha$-cusps, where $\alpha \in (0,1]$ 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.