I did say "diffusion-jump process". Look, even if we disregard jumps as a possible term in equations, merely considering volatility to be stochastic and driven

I did say "diffusion-jump process". Look, even if we disregard jumps as a possible term in equations, merely considering volatility to be stochastic and driven by a Brownian already gets you variances that may grow arbitrarily fast. This is without considering stranger nonlinearities on the Brownian term itself. People are overly impressed by forceful arguments of the Taleb variety and log-log plots of empirical distributions and start parroting talking points about heavy/fat tails. If I wasn't computerless and on my phone I would link to a paper that does the analog of the Anscombe quartet for log-log distribution plots "proving that data is Pareto/power law/etc." It's good vaccine for fat tail hipsterism, look it up.