The finiteness of moments of a stochastic exponential

It is well known that the stochastic exponential , of a continuous local martingale M has expectation EZt=1 and, thus, is a martingale if (Novikov's condition). We show that, for p>1, EZtp t} 0. As a consequence, we get that the moments of the stochastic exponential of a stochastic integral with respect to a Brownian motion are all finite, provided the integrand is a Brownian functional of linear growth.