Analysis of the Monte-Carlo error in a hybrid semi-Lagrangian scheme

We consider Monte-Carlo discretizations of partial differential equations based on a combination of semi-lagrangian schemes and probabilistic representations of the solutions. We study the Monte-Carlo error in a simple case, and show that under an anti-CFL condition on the time-step δt and on the mesh size δx and for N the number of realizations reasonably large, we control this error by a term of order O( √ δt/N). We also provide some numerical experiments to confirm the error estimate, and to expose some examples of equations which can be treated by the numerical method.