  We study a two-dimensional stochastic differential equation that has a unique
weak solution but no strong solution. We show that this SDE shares notable
properties with Tsirelson's example of a one-dimensional SDE with no strong
solution. In contrast to Tsirelson's equation, which has a non-Markovian drift,
we consider a strong Markov martingale with Markovian diffusion coefficient. We
show that there is no strong solution of the SDE and that the natural
filtration of the weak solution is generated by a Brownian motion. We also
discuss an application of our results to a stochastic control problem for
martingales with fixed quadratic variation in a radially symmetric environment.
