  We establish existence of exponential moments and the validity of the affine
transform formula for affine jump-diffusions with a general closed convex state
space. This extends known results for affine jump-diffusions with a canonical
state space. The key step is to prove the martingale property of an exponential
local martingale, using the well-posedness of the associated martingale
problem. By analytic extension we obtain the affine transform formula for
complex exponentials, in particular for the characteristic function. Our
results apply to a wide class of affine processes, including those with a
matrix-valued state space, which have recently gained interest in the
literature.
