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

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