Stochastic control problems for multidimensional martingales

We study a stochastic control problem for continuous multidimensional martingales, motivated by recent developments in robust finance and martingale optimal transport. In a radially symmetric environment, we explicitly construct the solution to this problem under mild regularity conditions. We consolidate some ideas from the theory of viscosity solutions of PDEs, which we then apply to solve our problem. Under a particular growth condition on the cost function, we solve the control problem in the two-dimensional case by proving that a weak solution of a certain SDE generates a Brownian filtration. We prove non-existence of strong solutions of this SDE and a related SDE, building on ideas from the study of Tsirelson's equation. These results lead us to conjecture that there is a gap between a Markov formulation of the control problem and a strong and weak formulation. Finally, we draw a connection to two further control problems. We characterise each of these problems in terms of viscosity solutions of a Monge-Ampère equation, similar to that which arises in the classical theory of optimal transport.