The setting of my question is the following. We have a diffusion process
$dS(t) = \mu S(t) \; dt + v(t,S(t)) \; dW(t)$
where $W$ is a standard Brownian motion under an equivalent martingale measure $Q$ and $v$ satisfies all necessary regularity conditions for there to exist a unique solution (i.e. linear growth, local Lipschitz etc.). The Linear growth condition is the one important for my question:
$v^2(t,y) \leq L(1+y^2) \text{ for every }t \in [0,T]$
I'm reading an article where it's claimed that due to the linear growth assumption the marginal density function $p_t(\cdot) = \frac{d}{dy}Q[S(t) \leq y]$ must have second moment i.e. $E[S^2(t)]<\infty$.
It's just mentioned in a footnote so I get the feeling that it's a well known result, but I cannot prove it! Any help is appreciated!
Thank you in advance