Hello
The key assumption here is that the return on the stock depends on the duration of time considered but not on the start time. These are denoted by u and t respectively in the example. We also denote the price of the stock at time t as \( S_t \) and the return from time t to time (t+u) as \( S_{t + u} / S_t \).
So, we are assuming that \( S_{t + u} / S_t \) does not depend on t. However, as written this doesn't look like an increment, which is what this section is all about. We can do a little manipulation to make an increment that looks similar by instead considering the log of the process:
let \( X_t = log(S_t) \) then we have \( X_{t+u} - X_{t} = log(S_{t+u}) - log(S_{t}) = log(S_{t+u} / S_{t}) \) (using log rules).
Now, if \( S_{t+u} / S_{t} \) does not depend on t (as per our assumption) then nor does \( log(S_{t+u} / S_{t} ) \).
So, our increments, \( X_{t+u} - X_{t} \) are stationary. This means that the distribution of \( X_{t+u} - X_{t} \) only depends on the duration u, not at what point in time the increment starts, t. So, any two increments of length u would have the same distribution, regardless of when they started.
I hope this helps
Andy
