Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 202 / 4 / b / Solution

For a continuous vector semimartingale $X$ and $f\in C^{1,2}$, the Itô formula isThe Itô product rule, or stochastic integration by parts, isThe additional second-order term is the quadratic covariation.

For general semimartingales with càdlàg paths, use left limits in the integrands. The Itô formula for semimartingales with jumps, for time-independent $f\in C^2$, isHere $X^c$ denotes the continuous local martingale part; the continuous bracket is that of these parts. For time-dependent $f\in C^{1,2}$ add $\int {\partial }_{s}f(s,X_{s-})ds$ and insert $s$ in the other evaluations of $f$. The integration by parts formula becomesFor continuous paths the jump sum vanishes, recovering the first two formulas.