Let $W\_t$ be a Wiener process with $W\_0=0$, and let $L=\{at+by=c\}$ be a line with $c/b<0$ (i.e. the line crosses the $Y$-axis below $0$).


Assume that $W\_t$ stayed above $L$ up to time $T$. What is the PDF of $W\_T$ under this assumption? Does it have a closed form?



Density of $W_t$ assuming it stayed above a line $L$
The problem can be restated as follows:



> 
> For a real $a>0$ and a real $b$, let
> $$X\_t:=a+bt+W\_t$$
> for real $t\ge0$, where $W$ is a standard Wiener process. Let
> $$\tau:=\inf\{t>0\colon X\_t=0\}.$$
> For a real $t>0$, find the joint distribution of $X\_t$ and $\tau$.
> 
> 
> 


The answer is well known:
$$P(X\_t\in dx,\tau>t)=f\_t(x-a-bt)(1-e^{-2ax/t})\,1(x>0)\,dx, \tag{1}\label{1}$$
where $f\_t$ is the pdf of $W\_t$. A derivation of this formula can be found [here](https://works.bepress.com/iosif-pinelis/24/).


