Let $X_t=\mu t + \sigma W_t$ where $B_t$ is the standard Brownian motion. What is the density for $X_{t^\prime}$ with $t^\prime \in [0,q]$ conditional on $X_t\geq 0 \ \forall t \in [0,q]$.
I tried starting the simple case with $\mu=0$ and $\sigma=1$. What I know is the distribution when $t^\prime=q$
$$f(x)=\frac{2(x)}{\sqrt{2 \pi q^3}} \exp \left(-\frac{x^{2}}{2 q}\right)$$
which follows from the joint distribution of the running minimum and value of of BM.
It would be great to get some tips about how to condition on the event of staying above 0 for the interval [0,q], when writing the density of of BM at time $t<q$.