Help me solve this problem:
Let $W_t$ be a Brownian motion and suppose $X_t = \int_{0}^{t}\delta _{s}dW_{s}$ where $\delta _{s}$ is a deterministic function. Then show that $X_t$ is a Gaussian process with mean, $m(t) = 0$ and covariance function $\rho (s,t)=\int_{0}^{min(s,t)}\delta _{s}^{2} ds$.
Edit: I am looking for a specific approach which utilizes Ito's Lemma and moment generating functions.