Let $\tau(x)$ be the first hitting time of a Lévy process $(X_t)_{t\geq 0}$ to level $x$. Let $f$ be a continuous function and $g(x)=\mathbb{E}[f(\tau(x))]$. Is it obviously true that $g$ is continuous?
2025-01-13 06:13:29.1736748809
Continuity with respect to hitting time level
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Doesn't seem to be right for Poisson process.