I would like to calculate the joint distribution of a Brownian motion with drift and an associated stopping time. But I am a bit lost. I want to find the distribution of the following:
$$\mathbb{P}\Big(\ U(1)\leq y\ ,\ \tau(b,U)\leq1\ \Big)$$
where
- $\tau(b,U):=\inf\{t\geq0:U(t)\geq b\}$ and
- $U(t):=vt+B(t)$
where $B(t)$ is the standard Brownian motion.
After some calculation I put the equation I wrote this as
$$\mathbb{P}\Big(\ B(1)\leq y-v\ ,\ \tau(b-vt,B)\leq1\ \Big)=\int_{\tau\leq 1}\mathbb{P}\Big(\ B(1)\leq y-v\ |\ \tau,B(\tau) \Big)$$
where I abbreviated $\tau:= \tau(b-vt,B)$
I am not sure if the last equality holds, nonetheless I don't know what to do afterwards. Any help is appreciated.
Suppose that $b>0$, denote the running maximum with $\overline{U}_{t}=\sup_{s\leq t}U_s$. Recall Brownian motion has (a.s.) continuous paths. Then, observe that $\tau(b,U)\leq t\iff \overline{U}_{t}\geq b$. It follows that $$P(U_t\leq y,\tau(b,U)\leq t)=P(U_t\leq y,\overline{U}_{t}\geq b)$$ The joint density of $(U_t,\overline{U}_{t})$ is known. The quantity you look for is $$P(U_t\leq y,\tau(b,U)\leq t)=\int_{(-\infty,y]}\int_{[b,\infty)}\underbrace{\frac{2(2m-w)}{t\sqrt{2\pi t}}e^{vw-\frac{v^2t}{2}-\frac{1}{2t}(2m-w)^2}}_{f_{\overline{U}_{t},U_t}(m,w)}dmdw$$ The density $f_{\overline{U}_{t},U_t}(m,w)$ is obtained from the driftless case one (see e.g. Karatzas, Shreve 8.1.) with a Girsanov theorem argument with the exponential martingale $Z_s=e^{-\frac{1}{2}v^2s-vB_s}=e^{\frac{1}{2}v^2s-vU_s},s\geq 0$ so that $dQ=Z_tdP$ where $U$ is a Brownian motion under the measure $Q$.