Importance sampling for a robust and efficient multilevel Monte Carlo estimator for stochastic reaction networks

The multilevel Monte Carlo (MLMC) method for continuous time Markov chains, first introduced by Anderson and Higham (2012), is a highly efficient simulation technique that can be used to estimate various statistical quantities for stochastic reaction networks (SRNs), and in particular for stochastic biological systems. Unfortunately, the robustness and performance of the multilevel method can be deteriorated due to the phenomenon of high kurtosis, observed at the deep levels of MLMC, which leads to inaccurate estimates for the sample variance. In this work, we address cases where the high-kurtosis phenomenon is due to \textit{catastrophic coupling} (characteristic of pure jump processes where coupled consecutive paths are identical in most of the simulations, while differences only appear in a very small proportion), and introduce a pathwise dependent importance sampling technique that improves the robustness and efficiency of the multilevel method. Our analysis, along with the conducted numerical experiments, demonstrates that our proposed method significantly reduces the kurtosis of the deep levels of MLMC, and also improves the strong convergence rate from $\beta=1$ for the standard case (without importance sampling), to $\beta=1+\delta$, where $0<\delta<1$ is a user-selected parameter in our importance sampling algorithm. Due to the complexity theorem of MLMC and given a pre-selected tolerance, $TOL$, this results in an improvement of the complexity from $\mathcal{O}\left(TOL^{-2} \log(TOL)^2\right)$ in the standard case to $\mathcal{O}\left(TOL^{-2}\right)$.