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 It\^o--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 $\epsilon > 0$, we prove that the method achieves a mean-square-error accuracy of $O(\epsilon^2)$ with a computational cost of $O(\epsilon^{-2} \big\vert \log \epsilon \big\vert^{3/2} (\log \big\vert \log \epsilon \big\vert)^{1/2})$ for uniformly Lipschitz continuous payoff functions and $O \big( \epsilon^{-2} \big\vert \log \epsilon \big\vert^{5/3 + \xi} \big)$ for discontinuous payoffs, respectively, where $\xi > 0$. 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.

翻译：本文针对漂移系数不满足压缩性条件的随机微分方程不变测度弱逼近问题，开发了一种高阶测度变换多层级蒙特卡罗(MLMC)方法。该方法通过采用1.5阶强伊藤-泰勒格式，在MLMC轨迹的成对耦合中引入弹簧项，从而在保持MLMC方法所需 telescoping sum 性质的同时，恢复漂移系数的压缩性。证明表明，对于任意时间T>0和充分小时间步长h>0，测度变换MLMC方法的方差随时间T线性增长。对于给定误差容限ε>0，证明该方法对一致Lipschitz连续支付函数实现O(ε²)均方误差精度时，计算复杂度为O(ε^{-2}|log ε|^{3/2}(log|log ε|)^{1/2})；对非连续支付函数则为O(ε^{-2}|log ε|^{5/3+ξ})，其中ξ>0。与M. Giles和W. Fang开创性工作中的Milstein测度变换方法相比，我们提出的高阶测度变换MLMC方法在计算复杂度相关的常数项上也有显著改进。通过多项数值实验验证了理论结果并评估了方法的鲁棒性。