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