Monte Carlo solution for the Poisson equation on the base of spherical processes with shifted centres

We consider a class of spherical processes rapidly converging to the boundary (so called Decentred Random Walks on Spheres or spherical processes with shifted centres) in comparison with the standard walk on spheres. The aim is to compare costs of the corresponding Monte Carlo estimates for the Poisson equation. Generally, these costs depend on the cost of simulation of one trajectory and on the variance of the estimate. It can be proved that for the Laplace equation the limit variance of the estimation doesn’t depend on the kind of spherical processes. Thus we have very effective estimator based on the decentred random walk on spheres. As for the Poisson equation, it can be shown that the variance is bounded by a constant independent of the kind of spherical processes (in standard form or with shifted centres). We use simulation for a simple model example to investigate variance behavior in more details.