# Paley-Wiener-Zygmund Integral

##### 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*}