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
\[ \phi(t) = \sum_{k=1}^N c_k \mathbf{1}_{[t_{k-1},t_k)}(t) \]
- 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
\[ \int_0^T \phi(t)^2 dt \] - * 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.
