Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 25 / 1 / c / 1 / Solution

Apply the construction of part (a) to the real Hilbert space $H=L^2([0,\infty ),du)$, and setChoose $X(0)=0$. The interval of length zero represents the zero vector of $H$, so $W_0=0$. All variables are defined on the single probability space already used for the isonormal Gaussian process.