The process is indeed stationary for all values of pi and O.
To answer your query, let's go through each part step-by-step.
a) For the mean function, we calculate it as an expected value of Xt. In other words, it is the mathematical expectation E[Xt]. Here, considering the fact that the expected value of noise Wt is zero, we have:
E[Xt] = E[pi + Wt+ Owt-1]
This equals to pi + E[Wt] + O * E[wt-1]. Given that the expected value of noise Wt is zero, the formula simplifies to:
μ(t) = pi
Hence, the mean function is μ(t) = pi.
b) The next step is to calculate the autocovariance function. We should find the covariance between Xt and Xt+h. Given that the noise is assumed independent over time, the expected value of the noise at different times is zero. This implies that only lag of 0 and 1 will produce non-zero autocovariance. At lag 0, we compare the term with itself, while at lag 1 we still have the noise from the previous term.
Accordingly, the autocovariance function γ(h) is computed as follows:
γ(h) = cov[Xt, Xt+h] = cov[pi + Wt+ Owt-1, pi + Wt+h+ Owt+h-1]
Therefore, the autocovariance is O^2 for h in [0,1], and 0 otherwise. Hence the autocovariance function is γ(h) = O^2 for h in [0,1], 0 otherwise.
c) Finally, we determine whether the process is stationary. A process is classified as stationary if its mean and autocovariance are independent of time t. From our calculations above, we observed that the mean function is a constant pi, meaning it doesn't depend on t. Furthermore, the autocovariance function γ(h) only depends on the lag h.