Bessel power local martingale ($X^{2-\delta }$)

For a Bessel process of dimension $\delta$, started at $x>0$, the Itô formula giveson its stochastic interval before hitting zero. Thus the power is a local martingale after localization away from zero. For $\delta \ge 2$ 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.

For Brownian motion started at radius $x>0$, the displayed process is a continuous local martingale. It is a bounded true martingale when $0<a\le x$, but a strict local martingale when $a=0$ or $a>x$. In the latter case the boundary is outside the starting point, so stopping does not remove the singularity at the origin. Brownian scaling gives $\mathbb{E}\Vert B_t{\Vert }^{2-d}=O(t^{-(d-2)/2})$; when $a>x$, the stopped expectation tends to $a^{2-d}<x^{2-d}$.

For a Bessel process of dimension $\delta >2$ started at $x>0$, its scale function of a one-dimensional diffusion gives ${\mathbb{P}}_x({\tau }_r<\infty )=(r/x{)}^{\delta -2}$ for $0<r<x$. In particular, ${\inf }_{t\ge 0}{X}_t>0$ almost surely. The minimum is attained because the process tends to infinity; the hitting formula alone already implies positivity of the infimum.