  An inequality for the $p$th power of the norm of a stochastic convolution
integral in a Hilbert space is proved. The inequality is stronger than
analogues inequalities in the Literature in the sense that it is pathwise and
not in expectation. An application of this inequality is provided for the
semilinear stochastic evolution equations with L\'evy noise and monotone
nonlinear drift. The existence and uniqueness of the mild solutions in $L^p$
for these equations is proved and a sufficient condition for exponential
asymptotic stability of the solutions is derived.
