  In mathematical Finance calculating the Greeks by Malliavin weights has
proved to be a numerically satisfactory procedure for finite-dimensional
It\^{o}-diffusions. The existence of Malliavin weights relies on absolute
continuity of laws of the projected diffusion process and a sufficiently
regular density. In this article we first prove results on absolute continuity
for laws of projected jump-diffusion processes in finite and infinite
dimensions, and a general result on the existence of Malliavin weights in
finite dimension. In both cases we assume H\"ormander conditions and hypotheses
on the invertibility of the so-called linkage operators. The purpose of this
article is to show that for the construction of numerical procedures for the
calculation of the Greeks in fairly general jump-diffusion cases one can
proceed as in a pure diffusion case. We also show how the given results apply
to infinite dimensional questions in mathematical Finance. There we start from
the Vasi\v{c}ek model, and add -- by pertaining no arbitrage -- a jump
diffusion component. We prove that we can obtain in this case an interest rate
model, where the law of any projection is absolutely continuous with respect to
Lebesgue measure on $\mathbb{R}^M $.
