= Solution
For continuous <semimartingales>, the <Stratonovich integral> adds half the <quadratic covariation> to the <Itô integral>:
$$
\boxed{S_t=I_t+\frac12[X,Y]_t.}
$$
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 $\frac12\sum\Delta X\Delta Y$; its limit is $\frac12[X,Y]$. 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 $\circ dY$.
Back to article page
