I am taking a course in financial mathematics(Ito-Integrals, Black-Scholes,...) and there is something that is not immediately clear to me. When constructing our stock price model, the integral $\int_0^t \sigma_s\;dW_s$, $\sigma_t$ being the volatility stochastic process, was the reason we had to construct the Ito-Integral. So we set up all the theory and here in my course notes it is stated that $\int_0^t \sigma_s\;dW_s$ really is an Ito-Integral which means that $E\big[\int_0^t\sigma_s^2 ds\big] < \infty$. So far so good.
Right now we are dealing with Ito-processes because our stock price model seems to be an Ito-process. According to my lecture notes for the stock price model to really be an Ito-process the volatility $\sigma_t$ needs to be square integrable meaning $\int_0^t\sigma_s^2 ds < \infty$. Is this normally the case? In my lecture notes it isn't stated anywhere so maybe this is a trivial implication from some of our assumptions which I overlooked.