  A well-known It\^o formula for finite dimensional processes, given in terms
of stochastic integrals with respect to Wiener processes and Poisson random
measures, is revisited and is revised. The revised formula, which corresponds
to the classical It\^o formula for semimartingales with jumps, is then used to
obtain a generalisation of an important infinite dimensional It\^o formula for
continuous semimartingales proved by Krylov to a class of $L_p$-valued jump
processes. This generalisation is motivated by applications in the theory of
stochastic PDEs.
