= Fourth-moment deficit and bracket variance identity
{title2=$\mathbb EX_t^4=3t^2-3\operatorname{Var}(\langle X\rangle_t)$}
The polynomial $X^4-6X^2\langle X\rangle+3\langle X\rangle^2$ has zero Itô drift. Finite fourth maximal and second bracket moments make it a true martingale. If $\mathbb EX_t^2=t$ and $\operatorname{Cov}(X_t^2,\langle X\rangle_t)=0$, then $\mathbb EX_t^4=3t^2-3\operatorname{Var}(\langle X\rangle_t)$. Equality at every time forces a deterministic clock and hence <Brownian motion> by the <Lévy characterization of Brownian motion>.
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