A dynamic programming principle for multiperiod control problems with bicausal constraints

We consider multiperiod stochastic control problems with non-parametric uncertainty on the underlying probabilistic model. We derive a new metric on the space of probability measures, called the adapted $(p,\mathrm{\infty })$–Wasserstein distance ${\mathcal{A}\mathcal{W}}_p^{\mathrm{\infty }}$ with the following properties: (1) the adapted $(p,\mathrm{\infty })$–Wasserstein distance generates a topology that guarantees continuity of stochastic control problems and (2) the corresponding ${\mathcal{A}\mathcal{W}}_p^{\mathrm{\infty }}$-distributionally robust optimization (DRO) problem can be computed via a dynamic programming principle involving one-step Wasserstein-DRO problems. If the cost function is semi-separable, then we further show that a minimax theorem holds, even though balls with respect to ${\mathcal{A}\mathcal{W}}_p^{\mathrm{\infty }}$ are neither convex nor compact in general. We also derive first-order sensitivity results.