What statistical tests allow for a comparison of the "roughness" of discrete functions? I have a data set of the revenues of 10 different large companies from the years 2000 through 2019. Here they are all plotted in one graph. The y-axis has a unit of billions of Euros:
                                                   
What I found interesting about this, is that the company with the brown revenue plot seems to have a very stable growth pattern. (This is the line with the least revenues in 2019; it consistently appears at the bottom from 2004 onwards.) In other words, the discrete function that maps the years to their corresponding revenues seems to be the least "rough" for the company with the brown colour.
We can formalize this idea of roughness by defining it as follows: for a vector $x$ containing the y-coordinates of the time series, the roughness $R$ is defined as
R = sd(diff(x))/abs(mean(diff(x))).

Here, sd denotes the standard deviation, diff comprises the vector of differences of the consecutive values of $x$, abs is the absolute value and the mean is the average.
Question: are there any statistical tests that allow one to compare the roughness of discrete functions with one another, and to conclude that one (set of) function(s) is indeed less or more rough than another (set of) function(s) ?
 A: One standard measure of roughness is volatility. This arises from transforming each series $X_t$ into $\log(X_t)-\log(X_{t-1})$, and taking the standard deviations of the results.
(Ordinarily the standard deviations $\sigma$ need to be annualized. So if we start with daily stock prices from years with $252$ trading days, the volatility is $\sqrt{252}\sigma$; if we start with quarterly revenue data, the volatility is $2\sigma$. For the data here, we don't need to annualize, but the data may be too small for significance.)
Now we can interpret the question

*

*Are two time series significantly different in roughness, as measured by volatility?

as

*

*Are the standard deviations of the transformed series significantly different?

or equivalently

*

*Are the variances of the transformed series significantly different?

We can answer this with an $F$-test, if the distribution of log-returns can be assumed normal, or with some of the other tests discussed at the same Wikipedia article.
As a side note: that lowest revenue line on the graph certainly looks less volatile than the others. Perhaps that company's revenue comes from long-term sales contracts, e.g. contracts signed in 2015 which determined a good amount of revenue for 2015-2019. In that case its revenue would look like an averaged version of the patterns from similar companies with shorter sales contracts.
