SDE and Stochastic calculus

Question

$W_t$ is 1 dimension Brownian morion.

$X_t=(cosW_t,sinW_t)$

Write SDE about $X_t$

I thought that $f(t,x)=(cosx,sinx)$, but I can't how "$t$" is expressed.

I heard that the hint of this question is to use Ito formula.

Answer 1

Outline: Use Ito's formula to show that $$ d(\cos W_t) = -\sin W_t \,dW_t - \frac{1}{2} \cos W_t \,dt$$ and $$ d(\sin W_t) = \cos W_t \,dW_t - \frac{1}{2} \sin W_t \,dt.$$ From here, using standard SDE notation, $$ dX_t = (-\sin W_t, \cos W_t) \,dW_t - \frac{1}{2}(\cos W_t, \sin W_t) \,dt$$ hence $$ dX_t = A X_t \,dW_t - \frac{1}{2} X_t\, dt$$ where $$A = \begin{pmatrix} 0 & -1\\ 1 & 0 \end{pmatrix}.$$

Can you fill in the details?