Let $\displaystyle Y(t)=\int_0^t v(s)ds+B(t)$, where $B(t)$ is the standard Brownian motion and $v(t)$ a deterministic function. Compute $m(t,y):=\mathbf E\Big[\max\limits_{s\in[0,t]} Y(s)\big|Y(t)=y\Big]$.
We easily can deduce these by the reflection principal and it turns out that the solution and $P(t,x,y):=\mathrm{Prob}\Big(\max\limits_{s\in[0,t]} Y(s)>x\big|Y(t)=y\Big)=e^{-2\frac{x(x-y)}{t}}$ is independent of $v$. However, I would like to know if we can formulate $m(t,y)$ directly as an initial-boundary value problem of the heat equation. Also I would like to know how we can deduce the independence of $P(t,x,y)$ on $v$ directly --- I deduced this by some cancellation via Cameron–Martin-Girsanov theorem.