# Linear-Boundary Crossing Problem for Brownian Motion

This is a question I came across while reading:

$W = (W_t)_{t\geq{0}}$ is a standard BM.

Let $\mu\in \mathbb{R}$, and let $\tau_{a}^{\mu}$ = $\inf(t>0;W_t = a + \mu{t})$ be the first passage time of a BM to the boundary $a+\mu{t}$.

I would like to know:

1. the Laplace transform of $\tau_{a}^{\mu}$;

2. the probability $\mathbb{P}(\tau_{a}^{\mu} < \infty)$.

I am thinking of using Girsanov Theorem to transform the BM. All solutions welcome.

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Question 2 has a straight forward solution using a differential equation approach: $\mathbb{P}(\tau^\mu_a<\infty)=1$ The following link (pp. 21 f.) explains it nicely (and is also very detailed) - could not write it much better. If you were to google "brownian motion linear boundary" you will get additional results.

Also if you are generally interested in this type of problem I can recommend the following paper on Integral Equations and the first passage time of BM. It contains a short literature review and deals with a more general boundry.

Question 1 mainly entails finding the density function for $\tau^\mu_a$ This density is also known as the Bachelier-Levy formula (also see here)

$p(t)=\frac{a}{t^{3/2}}\Phi(\frac{a+\mu t}{\sqrt{t}})$ with $\Phi(y)=\frac{1}{\sqrt{2\pi}}e^{-y^2/2t}$

Inserting this result into the general formula for the laplace transform gives: $\mathcal{L}(\tau^\mu_a)(s)=\mathbb{E}[e^{-s \tau^\mu_a}]=\int_{-\infty}^{+\infty}e^{-sx}p(x)dx$

The desired result then follows by straight forward integration.

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