= Knight theorem for orthogonal martingales
{c}
{title2=$M_j(t)=B_j(\langle M_j\rangle_t)$}
Let $M_1,\ldots,M_N$ be continuous <local martingales> starting at zero, with $\langle M_i,M_j\rangle=0$ for $i\ne j$ and $\langle M_j\rangle_\infty=\infty$ almost surely. The individual <Dambis-Dubins-Schwarz theorem> time changes produce <independent> standard <Brownian motions> $B_j$ with $M_j(t)=B_j(\langle M_j\rangle_t)$. Independence is between the <Brownian motions>; each may still depend on its own clock.
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