In this paper, we present a new antithetic multilevel Monte Carlo (MLMC) method for the estimation of expectations with respect to laws of diffusion processes that can be elliptic or hypo-elliptic. In particular, we consider the case where one has to resort to time discretization of the diffusion and numerical simulation of such schemes. Motivated by recent developments, we introduce a new MLMC estimator of expectations, which does not require simulation of intractable L\'evy areas but has a weak error of order 2 and achieves the optimal computational complexity. We then show how this approach can be used in the context of the filtering problem associated to partially observed diffusions with discrete time observations. We illustrate with numerical simulations that our new approaches provide efficiency gains for several problems relative to some existing methods.

翻译：本文提出了一种新的对跖多层次蒙特卡洛方法，用于估计涉及椭圆或次椭圆扩散过程律的期望。特别地，我们考虑了必须对扩散进行时间离散化并数值模拟此类方案的情形。受近期研究进展的启发，我们引入了一种新的多层次蒙特卡洛期望估计量，该估计量无需模拟难以处理的莱维面积，但具有二阶弱误差，并实现了最优计算复杂度。随后，我们展示了该方法如何应用于离散时间观测下部分观测扩散的滤波问题。数值模拟结果表明，与现有方法相比，我们的新方法在多个问题上提升了计算效率。