Ok, we’re at the very limits of my understanding, so don’t assume that this is exactly correct, but...
Take the risk-free rate, either how it’s standardly defined, or by just the size of the whole economy. The risk free rate is exponential, but that’s an artefact of it being a “rate”. You can have sub-exponential or super-exponential rates of growth of the whole economy, by varying the risk-free rate from year to year (or from moment to moment).
Then, in a well traded market, for reasons akin to what I mentioned above, every asset will be a random walk on the log scale, with the risk-free rate as the origin (ie if we continually adjust the values by the risk-free rate, we will get such a random walk).
Ok, we’re at the very limits of my understanding, so don’t assume that this is exactly correct, but...
Take the risk-free rate, either how it’s standardly defined, or by just the size of the whole economy. The risk free rate is exponential, but that’s an artefact of it being a “rate”. You can have sub-exponential or super-exponential rates of growth of the whole economy, by varying the risk-free rate from year to year (or from moment to moment).
Then, in a well traded market, for reasons akin to what I mentioned above, every asset will be a random walk on the log scale, with the risk-free rate as the origin (ie if we continually adjust the values by the risk-free rate, we will get such a random walk).
Ok, that seems consistent with what I said.