In this problem, we will show that the Ito integral of a deterministic function is a Gaussian Random Variable.

Let \(\phi\) be deterministic elementary functions. In other words there exists a  sequence of  real numbers \(\{c_k : k=1,2,\dots,N\}\) so that

\[ \sum_{k=1}^\infty c_k^2 < \infty\]

and there exists a partition

\[0=t_0 < t_1< t_2 <\cdots<t_N=T\]

so that

\[ \phi(t) = \sum_{k=1}^N c_k \mathbf{1}_{[t_{k-1},t_k)}(t) \]

1. Show that if \(W(t)\) is a standard brownian motion then the Ito integral
   \[ \int_0^T \phi(t) dW(t)\]
   is a Gaussian random variable with mean zero and variance
   \[ \int_0^T \phi(t)^2 dt \]
2. * Let \(f\colon [0,T] \rightarrow \mathbf R\) be a deterministic function such that
   \[\int_0^T f(t)^2 dt < \infty\]
   Then it can be shown that there exists a sequence of  deterministic elementary functions \(\phi_n\) as above such that
   \[\int_0^T (f(t)-\phi_n(t))^2 dt \rightarrow 0\qquad\text{as}\qquad n \rightarrow \infty\]
   Assuming this fact, let \(\psi_n\) be the characteristic function  of the random variable
   \[ \int_0^T \phi_n(t) dW(t)\]
   Show that for all \(\lambda \in \mathbf R\), show that
   \[ \lim_{n \rightarrow \infty} \psi_n(\lambda) = \exp \Big( -\frac{\lambda^2}2 \big( \int_0^T f(t)^2 dt \big)  \Big)\]
   Then use the the convergence result here to conclude that
   \[ \int_0^T f(t) dW(t)\]
   is a Gaussian Random Variable with mean zero and variance
   \[\int_0^T f(t)^2 dt \]
   by identifying the limit of the characteristic functions above.

   Note: When Probabilistic say the “characteristic function” of a random distribution they just mean the Fourier transform of the random variable. See here.