Multilevel Monte Carlo Simulation With Implicit Milstein Scheme

The general Euler scheme and Milstein scheme can diverge in strong and weak senses during simulating these SDEs with non-Lipschitz continuous coeffi-cients. Higham, Mao, Szpruch[Discrete and Continuous Dynamical Systems B, 2013, pp 2083-2100] proposed a new implicit Milstein discretization scheme in their article, and they prove this new scheme can preserve positivity for some kinds of SDEs with nonlinear coefficient. They also prove the strong convergence of this scheme under certain conditions, and give the proof of mean stability under some conditions.This article focuses on the A-stability analysis of this implicit Milstein scheme and try to explore the impact of parameter selection for the stability of this scheme. This article also gives the strong order of convergence for this method under global Lipschitz conditions and do some numerical test about comparing the computation cost between without using MLMC and using MLMC, and also test the difference between general Milstein scheme and this new implicit Milstein scheme under different stochastic models combining with MLMC method. This article also gives the comparison of computational cost of different parameters choices.