= Complex exponential of two orthogonal Brownian motions
{title2=$M_t=e^{B_t+i\vartheta_t}$}
For <orthogonal continuous local martingales> $B,\vartheta$ that are standard <Brownian motions>, the <real part> $X=e^B\cos\vartheta$ and <imaginary part> $Y=e^B\sin\vartheta$ are <continuous local martingales>. The <Itô formula> gives $dX=X\,dB-Y\,d\vartheta$ and $dY=Y\,dB+X\,d\vartheta$, since the two diagonal second-order terms cancel. Their <quadratic variations> equal $\int_0^te^{2B_s}ds$ and their <quadratic covariation> is zero. If $C=[B,\vartheta]$ is nonzero, the omitted drift terms are respectively $-Y\,dC$ and $X\,dC$, so the orthogonality hypothesis is essential.
Back to article page
