Thank you all, but I what I cannot understand is the real reason that Bellman equation is ususally solved by backwards. Can any one give an exmaple in which both intitial and terminal conditions are well defined, but the Bellman equation can only be solved by backwards?
I think there are tons of examples, the most proeminent being the backward heat equation [$]\partial_t u = -\Delta u[$], that fits Bellman framework AFAIR. The reason is that these kind of equations dissipate some entropy : you loose informations at each time, preventing you from going back forward to your initial state.
By the way sometimes the Bellman equations are forward, depends on example. For instance Burgers equations
are forward Bellman ones (to be precise, burgers equations fits the Jacobi-Bellman framework). Due to entropy dissipation, you can't move them backwards.