= Solution
For any initial point, rotate its vector through the Brownian angle:
$$
\boxed{X_t=x_0\cos B_t-y_0\sin B_t,\qquad Y_t=x_0\sin B_t+y_0\cos B_t.}
$$
This is a nonanticipating function of the prescribed Brownian path and has the required initial value. Equivalently, $Z_t=X_t+iY_t=(x_0+iy_0)e^{iB_t}$. The <Itô formula> gives $dZ_t=iZ_t\,dB_t-Z_t\,dt/2$, 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.
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