The multilevel Monte Carlo (MLMC) method for continuous-time Markov chains, first introduced by Anderson and Higham (SIAM Multiscal Model Simul 10(1):146–179, 2012), is a highly efficient simulation technique that can be used to estimate various statistical quantities for stochastic reaction networks, in particular for stochastic biological systems. Unfortunately, the robustness and performance of the multilevel method can be affected by the high kurtosis, a phenomenon observed at the deep levels of MLMC, which leads to inaccurate estimates of the sample variance. In this work, we address cases where the high-kurtosis phenomenon is due to catastrophic coupling (characteristic of pure jump processes where coupled consecutive paths are identical in most of the simulations, while differences only appear in a tiny proportion) and introduce a pathwise-dependent importance sampling (IS) technique that improves the robustness and efficiency of the multilevel method. Our theoretical results, along with the conducted numerical experiments, demonstrate 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 IS), to \(\beta =1+\delta \), where \(0<\delta <1\) is a user-selected parameter in our IS algorithm. Due to the complexity theorem of MLMC, and given a pre-selected tolerance, \(\text {TOL}\), this results in an improvement of the complexity from \({\mathcal {O}}\left( \text {TOL}^{-2} \log (\text {TOL})^2\right) \) in the standard case to \({\mathcal {O}}\left( \text {TOL}^{-2}\right) \), which is the optimal complexity of the MLMC estimator. We achieve all these improvements with a negligible additional cost since our IS algorithm is only applied a few times across each simulated path.

连续时间马尔可夫链的多级蒙特卡洛（MLMC）方法由Anderson和Higham首次提出（SIAM Multiscal Model Simul 10（1）：146–179，2012），是一种高效的仿真技术，可用于估算随机反应网络，尤其是随机生物系统的各种统计量。不幸的是，高峰度会影响多级方法的鲁棒性和性能，高峰度是在MLMC的深层次上观察到的现象，这导致样本方差的估计不准确。在这项工作中，我们处理高峰度现象是由于灾难性耦合导致的情况（纯跳跃过程的特性，在大多数模拟中，耦合的连续路径都是相同的，而差异仅以很小的比例出现），并引入了与路径相关的重要性抽样（IS）技术，该技术提高了多级方法的鲁棒性和效率。我们的理论结果以及进行的数值实验表明，我们提出的方法显着降低了MLMC深层的峰度，并且对于标准情况（无IS ），提高了\（\ beta = 1 \）的强收敛速度。）到\（\ beta = 1 + \ delta \），其中\（0 <\ delta <1 \）是我们的IS算法中用户选择的参数。由于MLMC的复杂性定理，并给出了预先选择的公差，\（\ text {TOL} \），从\（{\ mathcal {O}} \ left（\ text {TOL} ^ {-2} \ log（\ text {TOL}） ^ 2 \ right）\）在标准情况下为\（{\ mathcal {O}} \ left（\ text {TOL} ^ {-2} \ right）\），这是MLMC估计器的最佳复杂度。由于我们的IS算法仅在每个模拟路径上应用了几次，因此我们可以用很少的额外成本实现所有这些改进。