  We analyze the Snell envelope with path dependent multiplicative optimality
criteria. Especially for this case, we propose a variation of the Snell
envelope backward recursion which allows to extend some classical approxima-
tion schemes to the multiplicatively path dependent case. In this framework, we
propose an importance sampling particle approximation scheme based on a
specific change of measure, designed to concentrate the computational effort in
regions pointed out by the criteria. This new algorithm is theoritically
studied. We provide non asymptotic convergence estimates and prove that the
resulting estimator is high biased.
