Continuity of the probability that a Brownian motion with drift hits an upper barrier before the lower barrier in the drift Let $W$ be a Brownian motion and $u:\mathbb{R}_+ \to \mathbb{R}_+$ an upper barrier and $l:\mathbb{R}_+ \to \mathbb{R}_-$ a lower barrier.
Let $$\tau_u(\mu) = \inf\{ t \colon \mu t + W_t \geq u(t)\}$$ the first hitting time of the upper barrier and $$\tau_l(\mu) = \inf\{ t \colon \mu t + W_t \leq l(t)\}$$ the first hitting time of the lower barrier.
Define $p(\mu) = \mathbb{P}[\tau_u(\mu) \leq \tau_l(\mu)]$ as the probability that the Brownian motion with drift $\mu$ hits the upper barrier before the lower barrier.
I want to find conditions on $u,l$ such that $p$ is continuous (or differentiable).
 A: I encourage yourself to verify below computations. Please tell me in comments if I am wrong, so I can edit the errors.
Let $T>0$,
by Girsanov,
$$p_T(\mu) = \mathbb{P}\left[\tau_u(\mu)\leq \min(T,\pi(\mu))\right]=\mathbb{E}\left[e^{\mu W_T-\frac{\mu^2}{2}T}\mathbb{1}_{\tau_u(0)\leq \min(T,\pi(0))}\right]$$
assume (prove) $p_T\to_{T\to\infty} p$
then
$p_T$ is differentiable with respect to $\mu$, and again by Girsanov and using independence of increments:
$$\frac{dp_T}{d\mu}=\mathbb{E}\left[\left(W_T-\mu T\right)e^{\mu W_T-\frac{\mu^2}{2}T}\mathbb{1}_{\tau_u(0)\leq \min(T,\pi(0))}\right]=\mathbb{E}\left[W_T\mathbb{1}_{\tau_u(\mu)\leq \min(T,\pi(\mu))}\right]=\mathbb{E}\left[W_{\min(T,\tau_u(\mu))}\mathbb{1}_{\tau_u(\mu)\leq \min(T,\pi(\mu))}\right]$$
and you need to find a condition to have convergence when $T\to\infty$
for example, $l,u$ bounded and $u$ continuous should give by dominated convergence theorem:
$$\frac{dp_T}{d\mu}\to_{T\to\infty}\mathbb{E}\left[W_{\tau_u(\mu)}\mathbb{1}_{\tau_u(\mu)\leq \pi(\mu)}\right]=\mathbb{E}\left[\left(u(\tau_u(\mu))-\mu\tau_u(\mu)\right)\mathbb{1}_{\tau_u(\mu)\leq \pi(\mu)}\right]$$
but these are strong assumptions
