  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.

Probability