 Solution

For any initial point, rotate its vector through the Brownian angle:This is a nonanticipating function of the prescribed Brownian path and has the required initial value. Equivalently, $Z_t=X_t+i{Y}_t=(x_0+i{y}_0)e^{i{B}_t}$. The Itô formula gives $d{Z}_t=i{Z}_{t}d{B}_t-Z_{t}dt/2$, whose real and imaginary parts are exactly the equations in part c. Therefore the displayed formula is the strong solution of a stochastic differential equation, and uniqueness from part d makes it the only one.

The Brownian rotation in the plane preserves radius: $X_t^2+Y_t^2=x_0^2+y_0^2$. Nonzero starting points move around their initial circle, and the origin remains fixed.