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Let $f$ be the density of the stock asset under some model (Heston, SABR, Black Scholes, Variance-Gamma, etc).

Is $f$ square-integrable in these models?

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I've never heard about the density function of the stock price in Heston model. For pricing, one uses the characteristic function which can be derived from the Heston-PDE. In general, a density function is not always square-integrable. Check this post https://math.stackexchange.com/questions/756540/is-a-probability-density-function-necessarily-a-l2-function

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Yes, $f$ is square-integrate. This follows from the fact that the variance exists.

Note that in general for most models this is the case but it is debatable given that financial returns are so heavy tailed. Thus, by modeling finance with models that output returns with finite variance we might be heavily underestimating the risks.

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    $\begingroup$ Not sure of that: variance exists means that $S$ is square integrable, but that does not tell us for the density $f$. $\endgroup$
    – siou0107
    Dec 13, 2019 at 16:39
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    $\begingroup$ I agree with @siou0107, square-integrability is normally referred to the random variable, not its distribution function. $\endgroup$
    – Daneel Olivaw
    Dec 13, 2019 at 17:26

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