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.

-

1 Answer

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.

-
Wierd that you are posting links and saying my answer with links is not good cause links die :) –  adam Mar 3 at 10:38
@adam I also thought about that but my answer isn't links only. Even if the links died he could still use the answer for Question 1. Also for question 2 I named what to look for on google and privded the answer and mentioned that a PDE approach is necessary. You could still elaborate on Question 2 in detail in your answer to complement mine ;) –  Probilitator Mar 3 at 10:44