Let $X$ be a finite-variation Lévy process which starts at $X_0>0$, has positive drift, and has only downward jumps. Also define a stopping time $\tau := \inf(0\leq t \leq T: X_t<0)$, the first time $X$ crosses $0$.
I am trying to figure out the differentiability of the function defined as
$$ f(t, x) = \mathbb E[1_{\tau > T} \mid X_t = x] $$ in $[0, T]\times \mathbb R$.
My thoughts:
First of all, my intuition tells me that if $X$ has only finitely many jumps, then $f(t, x)$ has to be discontinuous at $x=0$, for the simple reason that $$ f(t, 0) \neq 0 \qquad \forall t\in[0, T] $$ while for $x<0$ $$ f(t, x) = 0 \qquad \forall t\in[0, T] $$ The case of infinite activity seems more complicated. I don't see that it is necessarily discontinuous at $x=0$ as it seems like $f$ is going to behave similarly to a case where $X$ is a diffusion: as it gets near $x=0$, $f(t, x)$ has to go to $0$. But then, what happens at $x=0$ and $t=T$?
So I am basically not sure how to carry out the analysis and would greatly appreciate if someone could shed some light on what steps I should take.