Higher-order spring-coupled multilevel Monte Carlo method for invariant measures
Sankarasubramanian Ragunathan, Håkon Andreas Hoel
TL;DR
The paper tackles the challenge of computing weak approximations of invariant measures for SDEs with non-contractive drifts by introducing a higher-order change-of-measure MLMC framework that uses a spring-coupled, order-1.5 Itô--Taylor scheme. Central contributions include rigorous $L^p$-stability and convergence results for the coupled fine/coarse paths, explicit bounds for the Radon–Nikodym derivatives, and sharp MSE–cost estimates distinguishing Lipschitz and discontinuous payoffs. The study demonstrates linear-in-time variance growth and achieves near-optimal computational costs: $\mathcal{O}(\epsilon^{-2} |\log \epsilon|^{3/2} (\log|\log \epsilon|)^{1/2})$ for Lipschitz payoffs and $\mathcal{O}(\epsilon^{-2} |\log \epsilon|^{5/3+\xi})$ for discontinuous ones, with experimental validation in 1D, 2D, and 3D systems. Overall, the method provides a robust, efficient tool for ergodic sampling in non-globally dissipative SDEs with wide potential applications in physics, biology, and beyond.
Abstract
A higher-order change-of-measure multilevel Monte Carlo (MLMC) method is developed for computing weak approximations of the invariant measures of SDE with drift coefficients that do not satisfy the contractivity condition. This is achieved by introducing a spring term in the pairwise coupling of the MLMC trajectories employing the order 1.5 strong Itô--Taylor method. Through this, we can recover the contractivity property of the drift coefficient while still retaining the telescoping sum property needed for implementing the MLMC method. We show that the variance of the change-of-measure MLMC method grows linearly in time $T$ for all $T > 0$, and for all sufficiently small timestep size $h > 0$. For a given error tolerance $ε> 0$, we prove that the method achieves a mean-square-error accuracy of $O(ε^2)$ with a computational cost of $O(ε^{-2} \big\vert \log ε\big\vert^{3/2} (\log \big\vert \log ε\big\vert)^{1/2})$ for uniformly Lipschitz continuous payoff functions and $O \big( ε^{-2} \big\vert \log ε\big\vert^{5/3 + ξ} \big)$ for discontinuous payoffs, respectively, where $ξ> 0$. We also observe an improvement in the constant associated with the computational cost of the higher-order change-of-measure MLMC method, marking an improvement over the Milstein change-of-measure method in the aforementioned seminal work by M. Giles and W. Fang. Several numerical tests were performed to verify the theoretical results and assess the robustness of the method.