This was part of an exam question.
Let $a<b$, $(X_t)$ a brownian motion and $$\forall x \in \mathbb R, t\ge 0, \quad \pi(x,t):=P(\forall s \in [0,t], x+X_s \in [a,b]).$$
Given that $\pi$ is $C^2$ (that was assumed in the question), I want to show that $\pi$ satisfies a partial differential equation of order $1$ in $t$ and $2$ in $x$ on $(a,b)\times(0, \infty)$ (I guess it will be the heat equation).
I tried to use Itō's formula without success.