In the case of white noise, if $\xi$ is deterministic, then
  \begin{align*}
  E\bigg|\iint_{\phi(U)}\xi(y,s)W(dy,ds)\bigg|^2
    &= E\bigg|\iint\xi(y,s)1_{\phi(U)}(y,s)W(dy,ds)\bigg|^2\\
  &= \iint|\xi(y,s)1_{\phi(U)}(y,s)|^2\,dy\,ds\\
  &= \iint_{\phi(U)}|\xi(y,s)|^2\,dy\,ds\\
  &= \iint_U |\xi(\phi(x,t))|^2\,|\det\phi|\,dx\,dt\\
  &= E\bigg|\iint_U \xi(\phi(x,t))\,|\det\phi|^{1/2}\,W(dx,dt)\bigg|^2.
  \end{align*}