Past exam of the continuous martingales and stochastic calculus course of the University of Oxford / 2023 / 1 / b / iv

Use the same finite truncation $T_m(h)={\sum }_{j\le m}⟨h,e_j⟩A_j$ for $f$ and $g$. Since $\mathbb{E}[A_i{A}_j]={\delta }_{ij}$ by independence, zero means and unit variances,Here ${\delta }_{ij}$ is the Kronecker delta. The Cauchy-Schwarz inequality and convergence in L2 justify passing to the limit in the product's expectation. The Parseval identity for a Hilbertian basis now givesBecause each $Wh$ has mean zero, this is also its covariance identity. Thus $W$ is an isonormal Gaussian process over $\mathcal{H}$.