  We introduce a framework for quantifying propagation of uncertainty arising
in a dynamic setting. Specifically, we define dynamic uncertainty sets designed
explicitly for discrete stochastic processes over a finite time horizon. These
dynamic uncertainty sets capture the uncertainty surrounding stochastic
processes and models, accounting for factors such as distributional ambiguity.
Examples of uncertainty sets include those induced by the Wasserstein distance
and f-divergences.
  We further define dynamic robust risk measures as the supremum of all
candidates' risks within the uncertainty set. In an axiomatic way, we discuss
conditions on the uncertainty sets that lead to well-known properties of
dynamic robust risk measures, such as convexity and coherence. Furthermore, we
discuss the necessary and sufficient properties of dynamic uncertainty sets
that lead to time-consistencies of dynamic robust risk measures. We find that
uncertainty sets stemming from f-divergences lead to strong time-consistency
while the Wasserstein distance results in a new time-consistent notion of weak
recursiveness. Moreover, we show that a dynamic robust risk measure is strong
time-consistent or weak recursive if and only if it admits a recursive
representation of one-step conditional robust risk measures arising from static
uncertainty sets.
