= Solution
{c}
For a twice continuously differentiable function, <Itô's formula> in this stopped diffusion reads
$$
f(Y_t)=f(1)+\int_0^{t\wedge\tau}f'(y_s)\,dW_s+
\frac12\int_0^{t\wedge\tau}\left(f''(y_s)+\frac{k-1}{y_s}f'(y_s)\right)ds.
$$
For $f(x)=x^{2-d}$,
$$
f'(x)=(2-d)x^{1-d},\qquad
\frac12\left(f''(x)+\frac{k-1}{x}f'(x)\right)=\frac{(2-d)(k-d)}2x^{-d}.
$$
Setting $d=k$ eliminates the drift. Moreover $f'$ is bounded on $[\alpha,\beta]$, so the stochastic integral is square-integrable on each finite horizon by the <Itô isometry>. Hence \b[$f(Y_t)$ is a true martingale], not merely a local one. The process itself is bounded on the stopped interval. For $k=2$, this particular function is the constant one; the logarithmic scale function will supply the nontrivial hitting identity.
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