  This paper studies distributionally robust optimization (DRO) in a dynamic
context. We consider a general penalized DRO problem with a causal
transport-type penalization. Such a penalization naturally captures the
information flow generated by the dynamic model. We derive a tractable dynamic
duality formula under mild conditions. Furthermore, we apply this duality
formula to address distributionally robust version of average value-at-risk,
stochastic control, and optimal stopping.
