  We study multistage distributionally robust linear optimization, where the
uncertainty set is defined as a ball of distribution centered at a scenario
tree using the nested distance. The resulting minimax problem is notoriously
difficult to solve due to its inherent non-convexity. In this paper, we
demonstrate that, under mild conditions, the robust risk evaluation of a given
policy can be expressed in an equivalent recursive form. Furthermore, assuming
stagewise independence, we derive equivalent dynamic programming reformulations
to find an optimal robust policy that is time-consistent and well-defined on
unseen sample paths. Our reformulations reconcile two modeling frameworks: the
multistage-static formulation (with nested distance) and the multistage-dynamic
formulation (with one-period Wasserstein distance). Moreover, we identify
tractable cases when the value functions can be computed efficiently using
convex optimization techniques.
