Stochastic integrals of nonrandom
$(l\times d)$-matrix-valued
functions or nonrandom real-valued
functions with respect
to an additive process
$X$ on $\mathbb{R}^d$ are studied. Here an additive process
means a stochastic process with independent increments, stochastically continuous,
starting at the origin, and having cadlag paths. A necessary and sufficient
condition for local integrability of matrix-valued functions is
given in terms of the Lévy--Khintchine
triplets of a factoring of $X$. For real-valued functions
explicit expressions of the condition are presented
for all semistable Lévy processes on $\mathbb{R}^d$
and some selfsimilar additive processes.
In the last part of the paper, existence
conditions for improper stochastic integrals $\int_0^{\infty-}f(s)dX_s$
and their extensions are given; the cases where $f(s)\asymp s^{\beta}
e^{-cs^{\alpha}}$ and where $f(s)$ is such that
$s=\int_{f(s)}^{\infty} u^{-2} e^{-u} du$ are analyzed.