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Let $M=\{M_t,\mathcal{F}_t;0\le t<+\infty\}$, $N=\{N_t,\mathcal{F}_t;0\le t<+\infty\}$ be two continuous local martingales with $M_0=N_0=0\text{ a.s.}$. If $\langle M\rangle=\langle N\rangle$, then could we say that $M$ and $N$ have the same distribution?

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No, consider Brownian motion $W_t$ and $$M_t=\frac{W_t^2-t}{2},$$ $$N_t = -M_t.$$ Source: slides by David Heath page 5.

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