# Wiener process exercise

I have problems with one exercise connected with Wiener process. It can be really important exercise for me but I don't have any idea how to do it and even how to start

We can define $\tau_a = \inf\{ t \geq 0 :X_t = a\}$

We have also $Y_a$ and $Y_b$ and these random variables are independent and they have the same distributions like $\tau_a$ and $\tau_b$. Show that $Y_a + Y_b \overset{d}{=} \tau_{a+b}$

Thank you for dedication your time.

• Here I think the main idea will be proving that the time from the moment when $a$ is first reached until the moment when $a+b$ is reached has the same distribution as the time until $b$ is first reached, and that time from first reaching $a$ until first reaching $a+b$ is independent of the time it takes to reach $a$ for the first time. – Michael Hardy Jan 20 '17 at 22:35
• You need $a$ and $b$ to have the same sign for the result to be true. Assuming that they do have the same sign you can use the strong Markov property to consider $X_{\tau_a+t}$ as a Brownian motion started at $a$. – user375366 Jan 21 '17 at 2:38