In your example simulating geometric random walk like process with fat tailed innovations, how do you know the mean exists?

In the following example in the MATLAB document: https://www.mathworks.com/help/econ/examples/using-extreme-value-theory-and-copulas-to-evaluate-market-risk.html

You model the asset returns as a process with heavy tailes. Particularly, the log differences are modeled by AR(1) process with Student t distributed innovations. Subsequently, the residuals are also modeled with GPD tails. You then obtain the exponent of the differences to get the forecast of asset prices.

How do you know that the forecast exists? For instance, there is no mean of exp(x) where x is from Student t. The tails are too heavy and so the integral doesn't converge, and the mean is undefined.