I'm working on a revision exercise that is as follows:
For Brownian motion B, find a function $f:\mathbb{R}\times \mathbb{R} \rightarrow \mathbb{R}$ such that
$$ L_t^0 (B) = \inf_{s \geq t} f(B_s, L_s^0(B)) $$
with probability one, where $L_t^0$ is the local time of the Brownian motion at zero.
I'm trying to understand where the complexity in this problem is - the local time $L_s^0 (B)$ is defined to be an increasing (or nondecreasing) process, and so I would expect that
$$\inf_{s \geq t} L_s^0(B) = L_t^0(B).$$
The only other difficulty could be if $B_t = 0$, but it can be shown that the local time is continuous (or admits a continuous modification) and so it would be equal to the right limit at $t$.