This article is concerned with the multilevel Monte Carlo (MLMC) methods for approximating expectations of some functions of the solution to the Heston 3/2-model from mathematical finance, which takes values in $(0, \infty)$ and possesses superlinearly growing drift and diffusion coefficients. To discretize the SDE model, a new Milstein-type scheme is proposed to produce independent sample paths. The proposed scheme can be explicitly solved and is positivity-preserving unconditionally, i.e., for any time step-size $h>0$. This positivity-preserving property for large discretization time steps is particularly desirable in the MLMC setting. Furthermore, a mean-square convergence rate of order one is proved in the non-globally Lipschitz regime, which is not trivial, as the diffusion coefficient grows super-linearly. The obtained order-one convergence in turn promises the desired relevant variance of the multilevel estimator and justifies the optimal complexity $\mathcal{O}(\epsilon^{-2})$ for the MLMC approach, where $\epsilon > 0$ is the required target accuracy. Numerical experiments are finally reported to confirm the theoretical findings.

翻译：本文研究多层蒙特卡洛（MLMC）方法，用于近似数学金融中Heston 3/2模型解函数的期望。该模型取值于$(0, \infty)$，具有超线性增长的漂移和扩散系数。为离散该随机微分方程模型，提出一种新的Milstein型格式以生成独立样本路径。所提格式可显式求解，且无条件保持正性，即对任意时间步长$h>0$均成立。这种大离散时间步长下的保正性质在MLMC框架中尤为可取。进一步，在非全局Lipschitz条件下证明了均方收敛阶为一阶，由于扩散系数呈超线性增长，该结果并非平凡。获得的一阶收敛性保证了多层估计量所需的相关方差，并验证了MLMC方法的最优复杂度$\mathcal{O}(\epsilon^{-2})$（其中$\epsilon > 0$为所需目标精度）。最后通过数值实验验证了理论结果。