Interchanging the order of differentiation and stochastic integration (Q801399)

Let \(\{M_ t,f_ t\}_{0\leq t\leq T}\) be a locally square-integrable martingale, and suppose that \(\{f(t,\theta,\omega)\}\) is a family of random variables, under some conditions, the authors prove that \(\frac{d}{d\theta} \int^{T}_{0}f(t,\theta,\omega)dM_ t = \int^{T}_{0} \frac{d}{d\theta} f(t,\theta,\omega)dM_ t\).