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If $\alpha < 1$, then $X$ has bounded variation and positive drift, and therefore it does not creep downwards; see Theorem 7.11 in Kyprianou's book Introductory Lectures on Fluctuations of Lévy Processes with Applications. This means that $$\mathbb{P}^x(X(\tau_{-1}) \ne -1) = 1,$$ for $x > -1$, where $$\tau_{-1} = \inf \{t > 0 : X(t) \leqslant -1\}.$$ In particular $$\mathbb{P}^x(X(\hat\zeta) \ne -1) = 1,$$ and consequently $$\mathbb{P}^x(\hat\zeta = \zeta) = 1.$$