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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.

Take a standard Brownian motion $B$ and defineThe Itô formula gives $N_t=2{\int }_0^{t\wedge 1}{B}_{s}d{B}_s$, so both are continuous martingales starting at zero. Their second moments are $\mathbb{E}{M}_t^2=t\wedge 1$ and $\mathbb{E}{N}_t^2=2(t\wedge 1{)}^2$, making them L2-bounded continuous martingales. At equal times, the odd moments of a centered normal random variable vanish, soThey are consequently weakly orthogonal continuous martingales by solution.

Their product, for $t<1$, is $B_t^3-t{B}_t$. The Itô formula givesThe continuous finite-variation term $2{\int }_0^t{B}_{s}ds$ is not identically zero: it has variance $4t^3/3$ at every $t>0$. Uniqueness of the continuous semimartingale decomposition therefore shows that $MN$ is not even a local martingale. Equivalently $[M,N{]}_t=2{\int }_0^{t\wedge 1}{B}_{s}ds$ is not zero, so the pair is not orthogonal continuous local martingales.

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.

For zero-starting L2-bounded continuous martingales, this condition is equivalent to $\mathbb{E}[M_t{N}_t]=0$ at equal times, by conditioning the later factor. The covariance identity for continuous square-integrable martingales also makes it equivalent to $\mathbb{E}[M,N{]}_t=0$. It is weaker than the vanishing bracket condition for orthogonal continuous local martingales: $B_{t\wedge 1}$ and $B_{t\wedge 1}^2-(t\wedge 1)$ have zero equal-time covariance but nonzero bracket $2{\int }_0^{t\wedge 1}{B}_{s}ds$.