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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 $dX=XdB-Yd\vartheta$ and $dY=YdB+Xd\vartheta$, since the two diagonal second-order terms cancel. Their quadratic variations equal ${\int }_0^t{e}^{2B_s}ds$ and their quadratic covariation is zero. If $C=[B,\vartheta ]$ is nonzero, the omitted drift terms are respectively $-YdC$ and $XdC$, so the orthogonality hypothesis is essential.

With the printed assumptions alone, the general quadratic covariation calculation from the two noise coefficients givesA continuous finite-variation process contributes no quadratic covariation, so the drift terms in the previous part do not alter these formulas. Taking $\vartheta =B$ gives $d[X,Y]=e^{2B}\cos (2B)dt$, which is nonzero near zero since $B_0=0$. Also $d([X]-[Y])=-4XYdt$ is not identically zero. Thus the requested assertions are false as printed.

Under the corrected assumption $C=0$, $X^2+Y^2=e^{2B}$ yieldsThe pair is therefore a pair of orthogonal continuous local martingales. This equality also identifies the clock for the intended time change of a continuous process.