Optimal control of martingales in a radially symmetric environment

We study a stochastic control problem for continuous multidimensional martingales with fixed quadratic variation. In a radially symmetric environment, we are able to find an explicit solution to the control problem and find an optimal strategy. We show that it is optimal to switch between two strategies, depending only on the radius of the controlled process. The optimal strategies correspond to purely radial and purely tangential motion. It is notable that the value function exhibits smooth fit even when switching to tangential motion, where the radius of the optimal process is deterministic. Under sufficient regularity on the cost function, we prove optimality via viscosity solutions of a Hamilton-Jacobi-Bellman equation. We extend the results to cost functions that may become infinite at the origin. Extra care is required to solve the control problem in this case, since it is not clear how to define the optimal strategy with deterministic radius at the origin. Our results generalise some problems recently considered in Stochastic Portfolio Theory and Martingale Optimal Transport.