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We study a two-dimensional stochastic differential equation that has a uniqueweak solution but no strong solution. We show that this SDE shares notableproperties with Tsirelson's example of a one-dimensional SDE with no strongsolution. In contrast to Tsirelson's equation, which has a non-Markovian drift,we consider a strong Markov martingale with Markovian diffusion coefficient. Weshow that there is no strong solution of the SDE and that the naturalfiltration of the weak solution is generated by a Brownian motion. We alsodiscuss an application of our results to a stochastic control problem formartingales with fixed quadratic variation in a radially symmetric environment.

SDEs with no strong solution arising from a problem of stochastic control