Moment bounds for solutions to SDEs

Question

I've been looking at Kuo's book on stochastic integration, and I noticed that in Section 10.7, he proves some estimates for solutions to SDEs, written as $$dX_t = f(t,X_t)dt + \sigma(t,X_t)dB_t$$. One of these results, Theorem 10.7.2, is a bound on the fourth moment of the solution in terms of the data and the time of integration: $$ E(|X_t|^4)\leq C(E|\xi|^4 + 1)e^{K(t-a)} $$ for a solution on the finite interval $[a,b]$. While I believe the result is correct, I'm a bit troubled by the derivation. Kuo obtains it by, essentially applying the Itô isometry and Jensen's inequality to the term $$ \left(\int_a^t \sigma(s, X_s)dB_s\right)^4. $$ Or, at least, that's how I'm reading the proof. What troubles me is that it would seem that you need to know $$ \int_a^t E|\sigma(s,X_s)|^4 ds < \infty $$ before the isometry can be applied. I believe this can all be justified by regularizing the problem the problem and passing to a limit, but I wanted to see if my reasoning is correct, or if there is some way to argue around this point.