Definition of stochastic integrals by integration by parts

In 1959, Paley, Wiener, and Zygmund gave a definition of the stochastic integral based on integration by parts. The resulting integral will agree with the Ito integral when both are defined. However the Ito integral will have a much large domain of definition. We will now follow the develop the integral as outlined by Paley, Wiener, and Zygmund:

1. Let \(f(t)\) be a deterministic function with \(f'(t)\) continuous. Prove that \begin{align*} \int_0^1 f(t)dW(t) = f(1)W(1) – \int_0^1 f'(t)W(t) dt\end{align*}
   where the first integral is the Ito integral and the last integral is defined path-wise as the standard Riemann integral since the integrands are a.s. continuous.
2. Now let \(f\) we as above with in addition \(f(1)=0\) and “define” the stochastic integral \(\int_0^1 f(t) * dW(t)\) by the relationship
   \begin{align*}
   \int_0^1 f(t) *dW(t) = – \int_0^1 f'(t) W(t) dt\;.
   \end{align*}
   Where the integral on the right hand side is the standard Riemann integral.

   If the condition \(f(1)=0\) seems unnatural to you, what this is really saying is that \(f\) is supported on \([0,1)\). In many ways it would be most natural to consider \(f\) on \([0,\infty)\) with compact support. Then \(f(\infty)=0\). We consider the unit interval for simplicity.

3. Show by direct calculation (not by the Ito isometry) that
   \begin{align*}
   \mathbf E \left[ \left(\int_0^1 f(t)* dW(t)\right)^2\right]=\int_0^1 f^2(t) dt\;,
   \end{align*}
   Paley, Wiener, and Zygmund then used this isometry to extend the integral to any deterministic function in \(L^2[0,1]\). This can be done since for any \(f \in L^2[0,1]\), one can find a sequence of deterministic functions in \(\phi_n \in C^1[0,1]\) with \(\phi_n(1)=0\) so that
   \begin{equation*}
   \int_0^1 (f(s) – \phi_n(s))^2ds \rightarrow 0 \text{ as } n \rightarrow 0\,.
   \end{equation*}