  In this paper, we compare the performance of two scenario-based numerical
methods to solve stochastic optimal control problems: scenario trees and
particles. The problem consists in finding strategies to control a dynamical
system perturbed by exogenous noises so as to minimize some expected cost along
a discrete and finite time horizon. We introduce the Mean Squared Error (MSE)
which is the expected $L^2$-distance between the strategy given by the
algorithm and the optimal strategy, as a performance indicator for the two
models. We study the behaviour of the MSE with respect to the number of
scenarios used for discretization. The first model, widely studied in the
Stochastic Programming community, consists in approximating the noise diffusion
using a scenario tree representation. On a numerical example, we observe that
the number of scenarios needed to obtain a given precision grows exponentially
with the time horizon. In that sense, our conclusion on scenario trees is
equivalent to the one in the work by Shapiro (2006) and has been widely noticed
by practitioners. However, in the second part, we show using the same example
that, by mixing Stochastic Programming and Dynamic Programming ideas, the
particle method described by Carpentier et al (2009) copes with this numerical
difficulty: the number of scenarios needed to obtain a given precision now does
not depend on the time horizon. Unfortunately, we also observe that serious
obstacles still arise from the system state space dimension.
