  We consider the probabilistic numerical scheme for fully nonlinear PDEs
suggested in \cite{cstv}, and show that it can be introduced naturally as a
combination of Monte Carlo and finite differences scheme without appealing to
the theory of backward stochastic differential equations. Our first main result
provides the convergence of the discrete-time approximation and derives a bound
on the discretization error in terms of the time step. An explicit
implementable scheme requires to approximate the conditional expectation
operators involved in the discretization. This induces a further Monte Carlo
error. Our second main result is to prove the convergence of the latter
approximation scheme, and to derive an upper bound on the approximation error.
Numerical experiments are performed for the approximation of the solution of
the mean curvature flow equation in dimensions two and three, and for two and
five-dimensional (plus time) fully-nonlinear Hamilton-Jacobi-Bellman equations
arising in the theory of portfolio optimization in financial mathematics.
