Given a stochastic differential equation, what is the quadratic variation?
e.g, assume $dX_t = X_t \cos{B_t}dB_t - \frac{1}{2}\sin^2{B_t}dt.$ Is this correct?
$$d\langle X \rangle_t = (dX_t)^2 = X_t^2 \cos^2{B_t}(dB_tdB_t) + \frac{1}{4}\sin^4{B_t}(dtdt) - X_t \cos{B_t}\sin^2{B_t}dB_tdt$$
And using the 'rules of multiplication': $dt \cdot dt = dt \cdot dB_t = 0, dB_t \cdot dB_t = dt $,
$$ d\langle X \rangle_t = X_t^2 \cos^2{B_t} dt $$