Multilevel Monte Carlo Method for Ergodic SDEs without Contractivity
For researchers computing expectations of ergodic SDEs, this method removes the contractivity requirement and significantly improves computational efficiency.
This paper introduces a multilevel Monte Carlo method for ergodic SDEs without contractivity, using a change of measure to achieve uniformly bounded strong error and variance that grows linearly in time, reducing computational cost from O(ε^{-3}|log ε|) to O(ε^{-2}|log ε|^2).
This paper proposes a new multilevel Monte Carlo (MLMC) method for the ergodic SDEs which do not satisfy the contractivity condition. By introducing the change of measure technique, we simulate the path with contractivity and add the Radon-Nykodim derivative to the estimator. We can show the strong error of the path is uniformly bounded with respect to $T.$ Moreover, the variance of the new level estimators increase linearly in $T,$ which is a great reduction compared with the exponential increase in standard MLMC. Then the total computational cost is reduced to $O(\varepsilon^{-2}|\log \varepsilon|^{2})$ from $O(\varepsilon^{-3}|\log \varepsilon|)$ of the standard Monte Carlo method. Numerical experiments support our analysis.