The affine transform formula for affine jump-diffusions with a general closed convex state space

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.