pretty new to this kind of maths, it's a little outside my usual area so would appreciate sense checking something.
I have read that the joint density function for Brownian motion and it's running supremum are: $$ \Bbb{P}(B_t \in dx, \sup_{0 \le s \le t}B_s \in dy) = \frac{2(2y-x)}{\sqrt{2\pi t^3}} \exp{\left(-\frac{(2y-x)^2}{2t}\right)} $$
I wanted to incorporate drift into this, so supposing that $X_t = B_t +ct$ for some $c \in \Bbb{R}$, I am trying to apply Girsanov's theorem here. I think I am right in saying that if I apply change in measure: $$ \frac{d\Bbb{Q}}{d\Bbb{P}} = \exp{(-cB_t - \frac{1}{2} c^2 t)} $$ Then $X_t$ is $\Bbb{Q}$-Brownian motion. So, if $\Bbb{1}$ is the ondicator function:
$$ \begin{align} \Bbb{P}(X_t \in dx, \inf_{0 \le s \le t}X_s \in dy) & = \Bbb{E}_{\Bbb{P}}\left(\Bbb{1}(X_t \in dx, \inf_{0 \le s \le t}X_s \in dy)\right) \\ & = \Bbb{E}_{\Bbb{Q}}\left(\left(\frac{d\Bbb{Q}}{d\Bbb{P}}\right)^{-1}\Bbb{1}(X_t \in dx, \inf_{0 \le s \le t}X_s \in dy)\right) \\ & = \Bbb{E}_{\Bbb{Q}}\left(\exp{\left(cB_t + \frac{1}{2} c^2t\right)}\Bbb{1}(X_t \in dx, \inf_{0 \le s \le t}X_s \in dy)\right) \\ & = \Bbb{E}_{\Bbb{Q}}\left(\exp{\left(cX_t - \frac{1}{2} c^2t\right)}\Bbb{1}(X_t \in dx, \inf_{0 \le s \le t}X_s \in dy)\right) \\ & = \exp(cx-\frac{1}{2}c^2t)\Bbb{Q}(X_t \in dx, \inf_{0 \le s \le t}X_s \in dy) \end{align}$$
Now since $X_t$ is $\Bbb{Q}$-Brownian motion we can use the above density function to get: $$ \Bbb{P}(X_t \in dx, \inf_{0 \le s \le t}X_s \in dy) = \exp\left(cx-\frac{1}{2}c^2t\right)\frac{2(2y-x)}{\sqrt{2\pi t^3}} \exp{\left(-\frac{(2y-x)^2}{2t}\right)} $$
I am not fully sure if this is correct. If it is, is this how results like this are usually derived? As I said, this is not my usual area of maths so would be interested in results and references in this area.