= Solution
{c}
Let $p=\mathbb P(y_\tau=\alpha)$. For $k\ne2$, the bounded stopped martingale from part (b) converges to $y_\tau^{2-k}$, so the <dominated convergence theorem> gives
$$
1=p\alpha^{2-k}+(1-p)\beta^{2-k}.
$$
Therefore the <boundary hitting probability from a diffusion scale function> is
$$
\boxed{p=\frac{1-\beta^{2-k}}{\alpha^{2-k}-\beta^{2-k}}\quad(k\ne2).}
$$
For $k=2$, applying <Itô's formula> to $\log Y_t$ gives a bounded stopped martingale instead, yielding
$$
\boxed{p=\frac{\log\beta}{\log\beta-\log\alpha}\quad(k=2).}
$$
If $k>2$ and $\beta>1$ is fixed, the first expression tends to zero as $\alpha\downarrow0$. A continuous path that reaches zero before it reaches $\beta$ must first reach every such $\alpha$ before $\beta$, so that event has probability zero. Any path hitting zero in finite time has a finite maximum up to that time, and therefore belongs to one of these zero-probability events with an integer upper boundary $\beta>1$. Their countable union still has probability zero. Thus \b[the process does not hit zero almost surely]. This is the relevant part of the <Hitting-zero classification for a Bessel process>.
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