= Exponential criterion for a continuous local martingale and its bracket
{title2=$e^{\theta X_t-\theta^2A_t/2}\text{ local martingales}\Longrightarrow [X]=A$}
Suppose $X,A$ are continuous, start at zero, and $A$ is increasing. If $e^{X-A/2}$ and $e^{-X-A/2}$ are <local martingales>, their logarithms first recover adaptation of $X,A$ and the <semimartingale> property of $X$. In the decomposition $X=N+V$, the two Itô drift measures give $dV+(d[N]-dA)/2=0$ and $-dV+(d[N]-dA)/2=0$. Thus $V=0$ and $[X]=A$. All real exponential parameters may be assumed, but these two suffice.
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