Some ideas for bias and variance reduction in the splitting algorithm for diffusion processes

In this article, we highlight a bias induce by the discretization of the sample Markov paths in the splitting algorithm. Consequently we propose to correct this bias using a deformation of the intermediate regions. Moreover, we propose two estimation methods of the optimal regions in the splitting algorithm to minimise the splitting variance. One is connected with partial differential equation approach, the other one is based on the discretization of the state space of the process.