= Bessel power local martingale
{c}
{title2=$X^{2-\delta}$}
For a <Bessel process> of dimension $\delta$, started at $x>0$, the <Itô formula> gives
$$
d(X_t^{2-\delta})=(2-\delta)X_t^{1-\delta}\,dW_t
$$
on its stochastic interval before hitting zero. Thus the power is a <local martingale> after localization away from zero. For $\delta\geq2$ this interval is the whole lifetime; at $\delta=2$ the power is constant. The unqualified assertion fails for a reflecting Bessel process of dimension one, which acquires <local time of a semimartingale> at zero.
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