A non-stationary $AR(1)$ process, which is a random walk, is constant in mean, but not constant in variance. How about the other $AR(p)$ processes with the order $p>1$? Are they constant in mean?
Solved – Is non-stationary AR(p) process constant in mean
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I suspect there is no general term that will cover all cases. Consider, for example, a white noise generator. In that case, we would just call it white noise. Now if the white noise comes from a natural source, e.g., AM radio band white noise, then it has effects including superimposed diurnal, seasonal, and sun-spot (11 year) solar variability, and man made primary and beat interference from radio broadcasts.
For example, the graph in the link mentioned by the OP looks like amplitude modulated white noise, almost like an earthquake. I personally would examine such a curve in the frequency and or phase domain, and describe it as an evolution of such in time because it would reveal a lot more about the signal structure by direct observation of how the amplitudes over a set of ranges of frequencies evolve in time with respect to detection limits as opposed to thinking about stationarity, mainly by reason of conceptual compactness. I understand the appeal of statistical testing. However, it would take umpteen tests and oodles of different criteria, as in the link, to incompletely describe an evolving frequency domain concept making the attempt at developing the concept of stationarity as a fundamental property seem rather confining. How does one go from that to Bode plotting, and phase plotting?
Having said that much, signal processing becomes more complicated when a "primary" violation of stationarity occurs; patient dies, signal stops, random walk continues, and so forth. Such processes are easier to describe as a non-stationarity than variously as an infinite sum of odd harmonics, or a decreasing to zero frequency. The OP complaint about not having much literature to document secondary stationarity is entirely reasonable; there does not seem to be complete agreement as to what even constitutes ordinary stationarity. For example, NIST claims that "A stationary process has the property that the mean, variance and autocorrelation structure do not change over time." Others on this site claim that "Autocorrelation doesn't cause non-stationarity," or using mixture distributions of RV's that "This process is clearly not stationary, but the autocorrelation is zero for all lags since the variables are independent." This is problematic because auto-non-correlation is typically "tacked-on" as an additional criterion of non-stationarity without much consideration given to how necessary and sufficient that is for defining a process. My advice on this would be first observe a process, and then to describe it, and to use phrases crouched in modifiers such as, "stationary/non-stationarity with respect to" as the alternative is to confuse many readers as to what is meant.
You did not specify which financial data you model. You can't make a blanket statement that all financial data follows random walk, it's simply not true. Certain series are known to look like random walk with a drift, e.g. some asset prices. Even that is a gross simplification.
Assuming that you're dealing with asset prices, it's common to deal with asset returns. For instance, if your asset price series are $p_t$, then you'd obtain the return series $r_t=p_t/p_{t-1}-1$, alternatively, the log return can be used $\Delta \ln p_t$. The returns are stationary series, e.g. a few thousand years ago the returns on loans were about the same as they are today.
You can try ARMA on ARMAX on returns, but these are constant conditional variance models. Asset returns are known to exhibit non-constant variance, even stochastic variance. Hence, models like GARCH are applied often times.
This is a big topic, and I'm only scratching the surface. Asset returns are largely unpredictable. There are certain cases when they are predictable, of course, e.g. long run returns are in some sense etc.
Best Answer
If we define non-stationary AR(p) processes as the ones having single unit root, they all can be written as
$$X_t=X_{t-1}+Z_t$$
where $Z_t$ is a stationary linear process. Then the answer is yes, they are constant in mean.
In general situation is more complicated even for the AR(1) process, since in general we can define it as
$$X_t=\mu+\rho X_{t-1}+Z_t$$
Now if $\rho=1$ the process is not constant in mean.