Past exam of the mathematics course of the University of Cambridge / 2017 / iii / Paper 202 / 2 / d / Solution

For continuous semimartingales, the Stratonovich integral adds half the quadratic covariation to the Itô integral:One can see the factor directly from symmetric endpoint sums. On a partition of an interval, replacing $X$ at the left endpoint by the average of its two endpoint values adds $\frac{1}{2}\sum \Delta X\Delta Y$; its limit is $\frac{1}{2}[X,Y]$. Thus the correction depends only on the continuous local martingale parts of the semimartingales. If either integrator or integrand has finite variation, the quadratic covariation vanishes and these two integrals coincide. Here the printed notation $\partial Y$ denotes Stratonovich integral integration, equivalently $\circ dY$.