Semimartingale integration by parts ($XY=X_0{Y}_0+X_{-}\cdot Y+Y_{-}\cdot X+[X,Y]$)

For càdlàg semimartingales, the product rule uses left-limit integrands and quadratic covariation including jump products. For continuous semimartingales it reduces to the Itô product rule without left-limit notation. This rule permits an induction proof of the Itô formula for polynomial functions.