I am considering an optimal stopping problem of the form: $$v(x,t)=\sup_{\tau}\mathbb{E}_{x}\left[g(x_{\tau})+a(t+\tau)\right]$$ This problem, with some requirements on $g$ and $x$, can be put into a form as in Optimal Stopping Rules (Shiryaev) by defining a new state space with $x'=(x,t)$ as our new state and letting $g'(x,t)=g(x)+a(t)$. The first entry characterization is true under some assumptions on $g$ and $x$, i.e. $$\tau=\inf\{t|v(x_t,t)=g'(x_t,t)\}$$
However, this result from Shiryaev requires (among other things) that $g'(x')$ be $C_0$ semi-continuous, which is defined as $P_{x'}(\liminf\limits_{t\downarrow 0}g'(x'_t) \ge g'(x'))=1$. This rules out (it seems, unless I am confused) some simple functions like $a(t)=\pmb{1}_{t\le 1}$. If, for instance, $g(x)$ is a constant, then the resulting $v(x,t)$ is not lower-semicontinuous, which is required for some results in Shiryaev (although $v(x,t)$ is left continuous with a downward discontinuity at $t=1$).
I am wondering if there are results out for alternative or weaker assumptions on $a$?