Thank you Alan. I can understand the option pricing problem. The BS pde must be solved by backwards, only because the terminal condition, i.e., the option pay off, is well defined. The initial option price is unknown(or cannot be difined), so need to be solved. This is not because of the data is non-stationary, isn't it? Actually, I think the data is stationary in option pricing problem.
Maybe I can propose a specific example: the market making problem. In this problem, I think the initial wealth or utility of market maker can be defined and the data should be stationary. However, as I read in most papers for this problem, Bellman equation or HJB PDE is only solved by backwards. An example is MARCO AVELLANEDA and SASHA STOIKOV's paper High-frequency trading in a limit order book.
So, in market making problem, can the Bellman quation be solved by forwards?
You can solve it backwards only if you know the model (transition probabilities & reward function). As ISayMoo wrote, you can always solve it forward. If you do not know the model, you simply sample the space of policies until you converge to the optimal one. It's less efficient
compared to backward induction in general (a confusing typo corrected).