A general affine Markov semigroup is formulated as the convolution of a homogeneous one with a skew convolution semigroup. We provide some sufficient conditions for the regularities of the homogeneous affine semigroup and the skew convolution semigroup. The corresponding affine Markov process is constructed as the strong solution of a system of stochastic equations with non-Lipschitz coefficients and Poisson-type integrals over some random sets. Based on this characterization, it is proved that the affine process arises naturally in a limit theorem for the difference of a pair of reactant processes in a catalytic branching system with immigration.
D. A. Dawson. Zenghu Li. "Skew convolution semigroups and affine Markov processes." Ann. Probab. 34 (3) 1103 - 1142, May 2006. https://doi.org/10.1214/009117905000000747