  Using the theory of stochastic integration developed recently by the authors,
in this article is proved an It\^{o} formula for Hilbert space-valued It\^{o}
processes defined with respect to a cylindrical-martingale valued measure. This
formula is applied to prove an integration by parts formula for It\^{o}
processes, to prove a Burkholder inequality for the stochastic integral, and
finally to prove an It\^{o} formula for Hilbert space-valued It\^{o} processes
defined with respect to a L\'evy process.
