 Solution

Define the quadratic covariation of two continuous semimartingales by polarization:The available quadratic variation convergence theorem and the polarization identity imply that the sums of products of simultaneous grid increments converge in uniform convergence on compacts in probability to $[X,Y]$. The quadratic covariation is a continuous finite-variation process, vanishing at time zero.

For a grid ending at $t_n=h_n\lfloor t/h_n\rfloor$, the elementary algebraic identitytelescopes. The first two sums converge by part (a), with the continuous processes $X,Y$ as integrands; continuous paths are left-continuous and locally bounded. The last sum converges to the quadratic covariation by the preceding polarization argument. Finally, $X_{t_n}{Y}_{t_n}\to X_t{Y}_t$ locally uniformly by continuity. HenceThis is the Itô product rule, or stochastic integration by parts. Uniqueness of the limit in uniform convergence on compacts in probability, followed by continuity, makes the identity hold simultaneously for all $t$ outside one null set. The proof used finite-grid algebra, part (a), and the permitted quadratic variation theorem; it did not use the Itô formula.