Past exam of the continuous martingales and stochastic calculus course of the University of Oxford / 2023 / 2 / b / ii / Solution

Apply the exact discrete product identityand sum over the deterministic partition of an interval. Since $B_0=0$,The left side converges in L2 space to ${\int }_0^{t}sd{B}_s$ by the definition and Itô isometry, sinceThe right-endpoint Riemann sum converges in L2 space to ${\int }_0^t{B}_{s}ds$: its squared error is at mostConsequently,As in (a)(iii), continuity extends fixed-time equality to equality of the continuous processes.