= Solution
For a continuous vector <semimartingale> $X$ and $f\in C^{1,2}$, the \b[<Itô formula>] is
$$
\boxed{f(t,X_t)=f(0,X_0)+\int_0^t\partial_s f(s,X_s)ds+\sum_i\int_0^t\partial_i f(s,X_s)dX_s^i+\frac12\sum_{i,j}\int_0^t\partial_{ij}f(s,X_s)d[X^i,X^j]_s.}
$$
The \b[<Itô product rule>], or stochastic <integration by parts>, is
$$
\boxed{X_tY_t=X_0Y_0+\int_0^tX_s\,dY_s+\int_0^tY_s\,dX_s+[X,Y]_t.}
$$
The additional second-order term is the <quadratic covariation>.
For general <semimartingales> with <càdlàg> paths, use left limits in the integrands. The \b[<Itô formula for semimartingales with jumps>], for time-independent $f\in C^2$, is
$$
f(X_t)=f(X_0)+\sum_i\int_0^t\partial_i f(X_{s-})dX_s^i+\frac12\sum_{i,j}\int_0^t\partial_{ij}f(X_{s-})d[X^{i,c},X^{j,c}]_s+\sum_{0<s\leq t}\left(f(X_s)-f(X_{s-})-\sum_i\partial_i f(X_{s-})\Delta X_s^i\right).
$$
Here $X^c$ 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 becomes
$$
X_tY_t=X_0Y_0+\int_0^tX_{s-}dY_s+\int_0^tY_{s-}dX_s+[X,Y]_t,\qquad [X,Y]_t=[X^c,Y^c]_t+\sum_{0<s\leq t}\Delta X_s\Delta Y_s.
$$
For continuous paths the jump sum vanishes, recovering the first two formulas.
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