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.

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Keywords Affine process, Catalytic branching process, Continuous state branching process, Immigration, Ornstein-Uhlenbeck process, Poisson random measure, Skew convolution semigroup, Stochastic integral equation
Persistent URL dx.doi.org/10.1214/009117905000000747
Journal Annals of Probability
Dawson, D.A, & Zenghu, L.I. (2006). Skew convolution semigroups and affine markov processes. Annals of Probability, 34(3), 1103–1142. doi:10.1214/009117905000000747