= Pathwise uniqueness for a multiplicative martingale equation
{title2=$dZ=Z\,dM,\quad Z_0=z>0\Longrightarrow Z_t=z\exp(M_t-M_0-\tfrac12[M]_t)$}
The <Itô formula> gives $d\log Z=dM-\frac12d[M]$ for a strictly positive solution. Therefore equal initial values give indistinguishable solutions. Conversely the displayed exponential satisfies the equation. This is uniqueness for a fixed continuous local martingale driver and does not require the exponential to be a true martingale.
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