For the first one, I get that second differences are stationary. Specifically if Vt is the time series of the investment value at time t, I get:
Vt = 100 * t * Yt, which is not stationary.
∇Vt = 100 * t * ∇Yt + 100Yt-1, which is still not stationary.
∇2Vt = 200 * (α + ϵ), which is stationary.
For Part B, I'm running into more trouble. Intuitively, the second difference is no longer stationary. It should have constant mean, but variance that grows with t as the increasing per share value increases the variance in Xt. Modeling this out in excel seems to support this.
Taking the log of the first difference and then taking the difference of that seems to get to a time series that is close to stationary. It looks like the result has constant mean and decreasing variance, but approaches constant variance as t grows.
If I specify the number of shares owned as St = St-1 + Xt, then I get:
∇Vt = St * Yt - St-1 * Yt-1 = St-1 * Yt + Xt * Yt - St-1 * Yt-1 = St-1 * ∇Yt + Xt * Yt
log(∇Vt) = log(St-1) + log(∇Yt) + log(Xt) + log(Yt)
∇(log(∇Vt)) = log(St-1 / St-2) + log(Yt / Yt-1) + log(∇Yt / ∇Yt-1) + log(Xt / Xt-1)
The first two terms should have means and variances that decrease towards 0, the second two terms I think should be stationary.
Am I on the right track? I haven't started on part C yet.